My great grandfather was apparently an illegitimate child, the oldest of the family, and it has been said that back in the day (1877, Scotland) it was common practice for such children to be raised by their fathers. Is this correct and what would be the reasoning behind it?
Interesting - I've just checked up in my copy of "Scottish Customs: From the Cradle to the Grave", by Margaret Bennett (Birlinn Ltd, 2004).
An interview with someone who emigrated from Scotland in the 1950s referred to what I thought standard practice would be (p.107) - the girl would have to go before the Kirk Session for public criticism. And there would be no public sympathy for her. Now I am sure that the normal position was that the man might also find himself in the same position - however, the interview in question describes the men getting off without any comeback or penalty. This clearly suggests they didn't have to bring the child up.
In the Annie Lennox episode of 'Who Do You Think You Are' (the UK version), an illegitimate child was definitely left with the mother.
But the interesting thing is that in the book, on p109, a 1774 document from Skye refers to the custom of handfasting. In this, it was the (alleged) custom of men of rank to take a year's trial of a wife. If mutual satisfaction followed, the marriage was deemed "good and lawful". But if either party insisted on a separation and a child had been conceived during that year, then the child was to be taken care of by the father and was to be ranked among his lawful children, next after his heirs. The child was not to be considered a bastard as the cohabitation was a custom intended to produce a good marriage.
I am as certain as I can be that the normal case was that the wrath of the Kirk Session would descend on both man and woman, with the man's subsequent contribution being a monetary one only. I am always slightly dubious about descriptions of customs like this handfasting and trial marriages, however, there does appear to be something at the root of this.
I would suggest that the "common" practice of the father raising the child, as described in the question was therefore not common at all, but that wherever and whenever handfasting took place then something very like the suggested practice did take place, giving rise to the suggestion.
In the realm of family history research, you will find a huge array of information that might become evidence for or against a claim. It can often be difficult to decide just how much weight to give to any particular piece.
Although this is a matter about which whole books have been written, there is simple rule of thumb upon which you can generally rely. Be VERY SCEPTICAL about any statement that begins with "It has been said that ..." or "Great grandmother always believed that ..."
The claims that follow such introductions are not always wrong, but should not be relied upon without locating independent documented corroboration.
In the case of your great grandfather, you seem to be describing the case of a boy in a family made up of his father, stepmother and half-siblings. There is certainly nothing strange in that. The best evidence of how his place in the family was regarded might be found in his father's Will.