It's unlikely that a family would refer to their own son as a nurse-child, but he might still be a blood relative. The linked question describes all sorts of reasons why someone might be a nurse-child, but if you have a confirmed DNA link to the adoptive family, that suggests a relationship of some sort.
I have an example of a nurse-child in a similar situation.
In the 1911 census he was listed as a "Nurse Child" and had a different surname to the family he was with. (He had their surname in all subsequent records, as do his descendants.) That led to a birth certificate. He was born in the workhouse with a mother named, but no father. His mother's surname turned out to be the same as his adoptive mother's maiden name, and I was able to demonstrate that they were sisters.
So he was raised by his aunt, and in this case his descendants would certainly have DNA matches to his adoptive family.
I think it's worth investigating the adoptive family further - look at their relatives (siblings, cousins...) and see if any registered the birth of a son at around the right time. His birth surname may well have been something other than Capon, which would explain why you haven't found his registration yet.