1901 was a puzzle as Burton, shown aged 12, born Rotherham was a stepson of a Thomas and Jane Rosamund Braddock.
But they married AFTER the census.
Marriages Mar 1902
Braddock Thomas Doncaster 9c 1131
Cook Ellen Doncaster 9c 1131
Hetherington William Henry Doncaster 9c 1131
Lamb Jane Rosemond Doncaster 9c 1131
so it IS your man, as it links well to the 1891 census. (couldnt see a death of Burton Snr tho up to 1901 - maybe they had to wait ? )
1901 RG13; Piece: 4409; Folio: 89; Page: 44 Bolton Upon Dearne, Goldthorpe. Barnsley Rd.
Thomas Braddock 50 Head of Household born Leics, Ravenstone: Colliery Clerk
Jane Rosemond Braddock 48 Wife born Lincs, Louth
Walter Thomas Braddock 13 son born Derby', Castle Donnington
Samuel Charles Braddock 17 son Nott's' Fiskerton
Maggie Braddock 11 dau Yorks, Darfield
Edith Olive Braddock 7 dau Yorks, Burton upon Dearne
Burton Lamb 12 Stepson Yorks, Rotherham
Christopher Eades 50 Boarder