At least one of the Canadian marriage records for James's daughter Alice on FamilySearch states that her mother's maiden name was Stains. Based on that I had a look at the English census returns and the birth registrations of the children mentioned in the censuses.
What I found was that a James Gray married Elizabeth Jane Staines in 1871 (a lot earlier than the profiles suggested) in the registration district of Romford, Essex. They had 14 children together over the next 25 years, before Elizabeth (Staines) Gray died in 1899.
Alice Jane (Gray) Wrigglesworth appears to have been one of those children, born in 1887, which is the year on her very confusing gravestone (1888-1979 is her husband's dates). (This is slightly older than the age given on her marriage certificate).
Meanwhile, according to the census returns, James remained a widower and went to live with one of his married daughters. He died in Edmonton, Middlesex in 1921.
On the face of it, this is all very straightforward. Which means I'm clearly missing something. Is this completely the wrong family? (eg did James move to Canada in 1911?).
You mentioned DNA in one of your change logs - does that tell you a different story? Is there something going on that the census returns didn't record?
ETA: the reason 13yo Alice wasn't in her father's household in the 1901 census is that she was working as an under-housemaid not very far away. (Also: Sources on profiles).