The fact that your ancestor married someone called Miller - a fairly common surname - has relatively little bearing on the probability of a Y-DNA test fingering him as the father. However, if the child never lived with his mother and her husband then I agree, it's highly unlikely that he was their son.
We do now have one useful clue - it seems quite possible that his father's name was Robert, which will help to narrow the field (even though it was twice as popular a forename in Scotland as in England), especially if a DNA test indicates that the father's surname was a common one. He may well not have known that he was illegitimate, and it's also quite possible that he wasn't aware that his mother was alive, or that he had any siblings - after all, there's no evidence that he ever lived with his mother and we all know how Home Children and other adoptees were treated. By the way, in the 19th century you could be classified as an orphan even if one parent was still living.
Opportunities to knock down illegitimacy 'brick walls' don't come along very often. I can appreciate that you may not be interesting in tracing the ancestry of a man who seems to have deserted your 2G grandmother rather than making an honest woman of her, but until you know who the father was it isn't possible to divine what the circumstances were (for example, he may have died - or been in the army and posted abroad).
Oddly enough, in these post-feminist days, I hadn't thought in terms of the mysterious GG GF abandoning my GG GM or otherwise Doin' Her Wrong, merely that she had a baby and didn't marry the father.
Peter, I had a private exchange with you about the naming of GGF's father on the marriage certificate and your comment at the time was "It's amazing how often people claim on their marriage certificate that their father bore the same name - and how often that's wrong."
Partly prompted by this comment, I have rather assumed that this Robert Campbell senior was complete fiction. The Campbell bit pretty definitely was, (or at least if true it was complete coincidence) From my brother's detailed search of records we KNOW that GGF wasn't born when he said he was on later documents, so an extra little fib to make himself seem legitimate would seem neither here nor there.
One thing my brother discovered was the day/month GGF gave for his birth was that of his mother's death. While this is coincidental, it looks significant. What we are postulating is that:
GG GM Jane Campbell had an illegitimate baby in 1856, married John Miller 1858. Baby Robert lived with the older 'relatives' for some unspecified period of time.
Some time before the 1871 census he was admitted to Mossbank Industrial Schools, Hogganfield under the name Miller, which suggests that his stepfather in some way acknowledged him -possibly he (or rather, his wife, Robert's mother) signed him in as being unruly and in need of reform.
In the course of 1871 or 1872 he made his way to England, in time to sign on a ship between Oct 1871 and March 1872. (Crew records run in 6-month blocks)
His mother died 11 December 1871 - at which point he would have been an orphan because she was his only legal parent. Quite possibly he left the school at this point, and in future aged himself by a few months (maybe for employment purposes) and said he was born 11 December 1855. (My brother tells me that if he had been admitted to the school on a voluntary basis (i.e. a parent had put him there, rather than the police) he might have been discharged on the death of his mother, his only parent.)
The links suggesting that he knew his mother and stepfather, possibly lived with them after the 1861 census are (1) the re-use of his mother's date of death (2) the indication that he was down under his stepfather's name in 1871 (and my brother couldn't find a likely Robert Campbell instead of this lad, or a likely Robert Miller in other censuses) and also (3) that the name Jessie was special to him, Jessie being a diminutive of Jane or Janet, which was the name of both his mother and a Miller daughter (his half-sister)who died 3 weeks short of her 2nd birthday.
I think he knew his mother and her family very well, and the circumstances of his birth, and that he re-invented himself when he came to England. I also think there was no love lost between him and his stepfather, and he felt no compunction in distancing himself, and saying that he was alone in the world. He seems to have been taken under the wing of an older shipmate, later marrying his daughter (my GGM), so perhaps he played the sympathy card at a very early stage,and then stuck with it.