Using DNA to identify a parent

Yes, my brother looked into the school
He wrote to me "Uncontrollable’ children could also be volunteered by their parents; in these circumstances parents could be asked to make a financial contribution (generally, they weren’t, it being found more trouble than it was worth).

and
"Leaving aside the deprivation of liberty, and the living conditions (Mossbank is alleged to have been cold and damp), the regime was intended to be remedial rather than reformatory or inherently punitive: education aimed to teach useful skills, and any labour the children undertook had to be useful and teach useful skills. (A report in the Glasgow Herald of the 1867 AGM of the Glasgow Industrial & Reformatory School Association says that the boys were taught shoemaking, tailoring, or paper bag making). Consequently an industrial school would give a better start in life than a poorhouse school, and there were mutterings about this: I now find in the Glasgow Herald 31 Jan 1872 an account of a meeting of a deputation seeking expansion of the Industrial Schools in London with Mr Bruce (the then Home Secretary) who said that about half the children admitted in the previous year should not have been, and there was nothing more demoralising for the poor than to see their neighbours’ children allowed to wander the streets neglected and then on those grounds taken in by the School Boards and lodged, clothed, fed and taught almost gratuitously (no mention of misrepresentation of uncontrollability , which various rate/taxpayers alleged; nor does he explicitly mention being on the side of ordinary hard-working families trying to do the best for their kids etc…). "

Re the mother's death date/ Robert's birth. I don't know why in particular he would not have known when he was born. Possibly he didn't actually have the certificate, but if we go along with the hypothesis that he was part of the family, he would likely have known he was born in July rather than December. When his mother died he was 15 not 16 - he added half a year to his age, where your model might have had him losing half a year. It does look, though, as if he adopted his mother's death date as his birthday. (And what is that you do with Shakespeare??)

Regarding Roberts in the family for whom he might be named - I'm not aware of any. His mother Jane Campbell's father was John, and her GFs were John Campbell and George Gordon. Her Gordon uncles were almost every conceivable name except Robert, and i don't have note of any Campbell uncles. Possibly Jane was sufficiently at odds with her family when the baby was born (she didn't give birth at home, but went to a house in a different area of Glasgow) that she was NOT naming him for any of them. She might have just liked the name. As you say, it is a fairly common one.

My brother has actually identified a Robert Campbell who *might* be the father - no relation to Jane, but brother of the wife of the couple who were looking after the boy in 1861. This Robert was born 1809 and was married at time of GGF's birth (which would be good reason for not marrying Jane). Although the investigation was based on the idea that 'if GGF said his father was Robert Campbell, why not believe him?', I guess this man is as likely a candidate as any....

 

As far as I am aware, Shakespeare's birth date and death date are the same. My wife's mother was born 13th July and died on 13th July...it does happen.

 

My tree is lousy with Roberts! Apart from one in every generation from this GGF (and possibly his father), my Norfolk family has lots. I mentally count through the generations as 'Robert who married Elizabeth', his son 'Robert who married Martha' then 'Robert who married Rebecca'.....