I think we can eliminate your Grandad, John William Thorpe being with close relatives of his mother, Mary Beatrice, in 1911.
Mary Beatrice's parents were William Thorpe and Mary Bowman
Marriage 04 Mar 1866, Newcastle All Saints
https://www.familysearch.org/ark:/61903/1:1:QLRN-NH3CThey had at least 3 other children, siblings to Mary Beatrice
Births 1866 Q4, Tynemouth District William Henry, mmn Bowman
Births 1869 Q2, Carlisle District, Jane Ellen, mmn Bowman
Births 1874 Q3 , Newcastle District, Margaret. mmn Bowman
Mary Thorpe (nee Bowman) died in 1902, aged 54
Newcastle Evening Chronicle 04 August 1902
page 2, col 1
THORPE Gateshead 25 North Tyne Street on 2nd inst. aged 54, Mary the dearly beloved wife of William Thorpe. Interment on Wednesday at Gateshead Cemetery; to lift at 3.15pm. Friends kindly invited.Burial Register Gateshead East 6th August 1902:
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSJ7-H9LQ-H?i=338&cat=8252771911 William Thorpe (Mary Beatrice's Dad) was recorded in Darlington - John William not with him
https://www.familysearch.org/ark:/61903/1:1:XWHC-L8N1911 William Henry (Mary Beatrice's brother) and his family - John William not with them
https://www.familysearch.org/ark:/61903/1:1:XWCJ-WBF1911 Jane Ellen (Mary Beatrice's sister) and family - John William not with them
https://www.familysearch.org/ark:/61903/1:1:XWMQ-GDZ1911 Margaret (Mary Beatrice's sister) and family - John William not with them
https://www.familysearch.org/ark:/61903/1:1:XWM6-S23JaSo not with immediate family :-( There is a chance he was with a friend and has been mistakenly entered on the census return as if he was part of their family.
Best suggestion I can come up with to try to learn more is, if possible, visit Tyne and Wear Archives and look at the records for the workhouse in Newcastle. (their opening times are quite restricted so do check first)
Hopefully they will tell you where Mary Beatrice had been living when she was admitted and the reason she was in the hospital. The archives staff may be able to advise where, if John William had been in the care of the authorities rather than with a friend or relative, you may look to try to find a record for him.
Let us know when you get Granda's birth cert and tell us where he was born, that may give some clues.
Really sorry not to have found him, I couldn't find my Granda in 1911 either and as he was the very first person I searched for when it was released it drove me crazy! He was born in 1885 and, quite by chance, many years later I discovered he had gone to Canada for about 18 months! Quite why he'd never mentioned that when I was young remains a mystery.
Good luck with the searches, its great you have contact with extended family.
Boo