The new GRO indices make it much easier to construct complete birth families without spending a lot of money on certificates, but, as a favourite teacher once said: it may be necessary but it's not sufficient. Relying only on the new GRO indices will lead to errors, as the OP's self-answer shows; you must cross-correlate them with census, marriage and death records.
I've been doing a lot of work recently trying to track the trajectory through life of 332 individuals born with the same or similar name in England and Wales between 1860 and 1870. (Not relevant to this answer, but I'm attempting to eliminate most of these as an ancestor as more conventional methods have left the question wide open, and other researchers across the 'net have come to wildly different conclusions, most of which are obviously flawed once you start looking at the full data-set in detail).
The method I've been using to construct birth families is as follows. It can be time consuming, but it yields good results. It assumes you know the name, and (perhaps) year of birth registration for one child in your family of interest (hereafter called the Known Child.) In most scenarios, this will be your ancestor and you will already have a birth certificate that gives you the information you need, or be working from a marriage record or census information (although these are less reliable and require you to cast your net wider). If you don't have a Known Child, you may wish to spend some time determining a starting point by conventional means, but the brute force approach that I've been following is also an option.
First, a general caveat about searching for forenames and surnames: do not assume that they were recorded correctly and consistently in original sources, or that they've been transcribed correctly. And don't assume that different search engines generate identical variants to search -- they don't. Work out a likely set of variants for surnames (including common mis-transcriptions) and search for them all explicitly, in all the places you search (ditto for forenames). The GRO indices have some provision for finding variations but in my experience they're not superb -- they either bring up far too many alternatives that are unlikely or they miss some obvious ones. I've seen Craigne not find Craine or Crane, and Rathbond not find Rathbone. Other search engines have their own foibles. At least if you search explicitly for each variant you can think of, you will know what has been covered; relying solely on 'black box' search engines leaves you in the dark about what has actually been searched. How can I identify all the possible alternatives for a surname and other answers to the same question suggest ways you might generate a set of variants for surnames; and Is there a standard reference for alternative forms of a given or personal name? addresses forenames.
Second, another caveat about locations: keep an open mind about the location of birth registrations/census entries/marriages and deaths for members of the same family. If you're lucky they'll stay in a single or adjacent registration districts, but I've seen families that criss-crossed the country having children -- including children that were born and died between census so you might never have known the family moved away from 'home' for a period. So don't discard any entries until you've cross-checked all the data. It's helpful to have a good idea of the registration 'geography' in the areas where you find candidate children and GenUKI is an excellent resource for this.
Third, bear in mind that children may not have had their birth registered at all before 1874 when it became compulsory to register a birth, and to do so within 6 weeks. Before 1874 birth registration was not compulsory (unless the registrar demanded the details, which was a bit catch-22 -- how would the registrar know to ask if you hadn't already told him) and there was no time limit and penalties. Deaths were very likely to be recorded (as a death registration was a necessary prerequisite for legal burial); and as marriages were recorded by the person performing the ceremony, they were most likely of all to be recorded.
However, the process of transferring marriage information from (perhaps) a church to the local registrar, and from there to the GRO introduced opportunities for loss or corruption of data. Likewise, death and birth information had to go from the local registrar to the GRO and be indexed there (once when it was first received, and very recently to produce the new GRO indices). Both sets of indices are flawed (the new GRO indices are missing birth entries for the December quarter in 1860 in district 6c, for example, and the old indices and modern search sites based upon them have duplicates and mis-transcriptions as well as missing entries) so it's worth cross-checking entries you find in the new GRO indices with the old indices (access via FreeBMD) and with local bmd indices (if they exist) produced by volunteers from the records still held in local register offices (try UK BMD). These may not have mother's maiden surnames or age of death like the new indices, but the cross-check can alert you to inconsistencies and missing entries.
And a final caveat: do not assume that there was only a single family with the same surname and mother's maiden name in an area, having children in the same timeframe. Sometimes you're lucky and have a pair of unusual surnames to work with, but even so, don't discount brothers marrying sisters or cousins marrying cousins. Assume all the children you find come from different families until you have enough evidence from other sources to confirm or refute family groupings.
So, the process...
Note the Known Child's parents names and her/his place and date of birth. (If you don't have anything other than a birth name and an approximate date of birth, you have more work to do as you have to investigate all the possible mother's maiden names within a particular timeframe rather than narrowing the set of children down to a single maiden name at the start.)
Search the GRO indices for possible siblings (same surname, same mother's maiden name) for 20 years either side of the birth of the Known Child. This allows for the Known Child being the youngest, oldest or somewhere in the middle of the sequence of children in this marriage -- at this point you may miss siblings (less likely given you're looking at a 40 year span, which is longer than most women's childbearing time, but it still might be the wrong 40-year span!). However if you have missed any, you'll go back and mop them up later. Make a note of every entry you find -- cut and paste into a text document or spreadsheet. (A spreadsheet is better as it allows more flexibility for separating data into columns and then annotating and manipulating it).
Be systematic: search in 5 years bands (year +/- 2 years) by female and then male gender and keep track of what you've done -- it's easy to miss a band of years and/or a gender if you're not systematic, plus you won't want to repeat searches you've already done when you go back to mop up.
Cross-check what you find with other sources of birth registration data, primarily to identify possible gaps in the new GRO indices.
Now the fun begins. You can make a tentative start on grouping the children into 'pools' based on date and place of birth registration. At this stage it can only be tentative, so be prepared to rethink. However, children registered less than 6 months apart are unlikely (but not impossible) to be siblings -- remember there was no time limit for registering births before 1874; ditto children registered in the same year and quarter in different districts. Children with a common element in their name are probable siblings (for example, if you find a sequence of children with the middle name Willan). And children born in the same year quarter and district with identical or sequential GRO references may be twins (or it might just be a coincidence). Many families in the nineteenth century seem to have children spaced at 18 -24 months, but of course this doesn't allow for miscarriage or stillbirth (which didn't have to be registered before 1874), or illness of the mother; and families with children born every year are not uncommon either. Some children died before their birth was registered, so the only evidence for their existence may be a death registration.
Make a note of your tentative pools (including the 'pools' of one that you as yet have no reason to link together).
Now, cross-correlate the birth entries with census entries. I check the first census after the Known Child's birth and each census after that. (If the list of children you've found includes children born before previous censuses, include those too). Although it might be tempting to search for the Known Child, I try to pick the children in each pool with less common names (try Ilderton Smith rather than Mary Smith, for example) and/or those born closest to the census date (more likely although not certain to be in the same geographical region); and use the information about the family for that individual to refine the likely pools of siblings and match them to parents' names.
Cross-check their age and place of birth from the census with the birth registration details, to strengthen the identification of each one, although this isn't conclusive: I've seen a child of 11 months indexed as 11 years old and vice versa; try to review the actual census images if you have access, to spot transcription errors (although this won't help if the transcription error was made when the census schedules were compiled). And don't believe ages and places of birth -- it might be true that a mother never forgets the date and place of a child's birth, but who's to say the person filling in the census was the mother or consulted the mother. What you're looking for at this stage is a sequence of children of approximately the right ages and names (or name variants) to refine and correct your 'pools'. Bear in mind that children born shortly before a census may not be registered until after that census (so show up 'early').
Make a note of all the children you find in each family, and also of other people present who can shed light on the identification of the family (fathers-in-law, wife's sisters... anything that confirms the likely maiden name of the wife). You will often find children you didn't expect in the census(es): consider that (1) your preliminary search may not have started early enough or finished late enough to locate all the births in a particular family; (2) a wife may have remarried after the death of a previous spouse and brought step-children into the family who aren't necessarily identified as such and may be recorded with the 'new' surname; (3) a husband may have remarried and had children previously or subsequently registered under a different maiden name to the Known Child. (You may also encounter a total mess, such as the family of Joseph Harper who initially had children with Sarah Dainty and subsequently with Maria Trowman, who also had children from a previous marriage to an [unknown] Russ.. All the children are shown in the census after the remarriage with the surname Russ, no matter who the mother was, and there are two girls in the family shown as Mary Russ in the census born in the same year. Haven't quite sorted that one out yet -- at the next census, both girls are missing).
And there will always be children you search for that you cannot find anywhere -- not with parents, not with grandparents, not in institutions...
Do the same with emigration records that can also help group families together.
You should now have (A) a set of families with one or more named parents, and a mix of expected and unexpected children; (B) a set of individual children who can't be conclusively linked to those families (little Johnny with his paternal grandparents in London, and more than one family he might belong to); and (C) a set of children missing in one or more censuses (which might look like entire families)..
Taking (C) the missing children first, you'll probably be familiar with the techniques for tracking them down. Are they old enough to have married and set up home elsewhere? Or gone into service, or otherwise moved away to find work? Did they or their whole family emigrate? Did the mother remarry and the children show up in the census under a different surname?
One possibility now open to you with the new GRO indices is searching for deaths with an age at death. However, I recommend not using the age at death as a search criteria: (1) if you're searching in 5 year bands, the age of death isn't useful; and (2) the indices have indexed ages in months as ages in years! So someone shown as 5 at their death might actually be 5 months. Search for the name (and gender) of the missing child in five years bands up until the latest census you'd expect them to appear in with their family. Consider how the place of registration of the death corrrelates with the place of registration of the birth and of births either side of that in the putative family/families and the locations known from censuses. Unusal names (e.g. Mary Adey Harper) will leap out at you; for others, you might have a provisional hypothesis of which child it is, but you'll never be certain uless you can find a smoking gun in the form of a burial register entry with identifying information (or acquire the death certificate).
Now for (B) the lone children. They might show up in the bosom of one of your target families in a subsequent census, perhaps with one or both parents (or step-parents) or lodging with a sibling. Or they might show up in a completely different family (and thus turn out to be a red herring). Or, by a process of elimination, you might end up with only one place they can fit into the jigsaw based on the data available to you at the time. Again, you'll never be certain without a birth certificate or baptism and ideally other corroborating information, and there will always be children who appear or vanish seemingly from or into nowhere.
Finally (A) the set of families with one or more named parents and a mix of unexpected and expected children.
First, search for likely marriages of the parents -- cast the search net as widely as possible to include parish registers/BTs and local BMD indices as well the GRO indices (PRs and BTs and local indices can be more specific about a pairing than the GRO indices, which often only offers a set of possible combinations -- marriages have not yet been re-indexed by the GRO). The usual subscription sites are useful for this (FindMyPast, Ancestry, TheGenealogist, MyHeritage) as is FamilySearch and other free sites like those maintained by online parish clerks. [These sites might also help with the baptism of 'lone' children, or burial of missing ones].
If you've got some unexpected children, search as well for remarriages of either parent. Remember a woman who remarried may be indexed in her maiden or previous married name, depending on the guidelines given to the people doing indexing/transcription.
If you find a possible remarriage, look backwards for the two parts of the family, and check the GRO indices for births from the new pairing and/or previous pairings, to confirm that you're looking at the correct 'melded' family.
You'll now be building up a very good picture of the trajectories and membership of (hopefully) a small set of families, one of which is your family of interest (identified by the includion of your Known Child); and also have a list of 'strays' that don't fit in anywhere. You'll have sound supporting data for many of your conclusions about parents and siblings that belong together (and as important, those that don't). (At this point looking at online trees might clear up some lingering questions -- or it may raise your blood pressure because of the errors you spot. Never take them for gospel, and always check the conclusions against your own collected data and/or new searches that they suggest, but they can point you to records you may have missed, as a last resort).
In an ideal world, you'll have a 'perfect' family with no possible strays that might or might not belong and you can document your results with a sigh of relief. More likely you'll have a set of 'definites' and some 'possibles' and you'll have to decide how to document the 'possibles'. I make a note that these 'possibles' may be members of the family but it isn't proven one way or the other; and will probably return to them periodically to see if new records have become available that help slot them where they belong.
Edited to add: Useful reference material about civil registration: Ancestral Trails, by Mark D. Herber and Society of Genealogists