It's a good call. In the BBC programmes I referred to earlier, they showed a clip of Sheffield United featuring the legendary goalkeeper 'Fatty' Foulke. This clip is on the BFI site (
http://player.bfi.org.uk/film/watch-sheffield-united-v-bury-1902-1902/). The goalkeeper in the City clip is definitely not Fatty Foulke.
Fatty Foulke signed for Chelsea from Sheff U in May 1905 (according to
http://spartacus-educational.com/SHEFFUfoulke.htm).
So, if the City clip is the 1904-05 season, we would expect to see Fatty Foulke in goal.
If you have never heard of Fatty Foulke, he is worth reading up on. 6' 4", weighing at one point 24 stone, he is believed to be the butt of the question 'Who ate all the pies?' He reputedly once said about the fans: "I don't mind what they call me as long as they don't call me late for my lunch." And in November, 1904, when Bury beat Sheff U 7-1, a newspaper report stated: "Foulke's weakness is his inability to capture low shots. He got low ones on Saturday."
Best of all, Wikipedia reports that: "At the end of the first match in the 1902 Cup Final Foulke protested to the officials that Southampton's equalising goal should not have been allowed. Foulke left his dressing room unclothed and pursued the referee, Tom Kirkham, who took refuge in a broom cupboard. Foulke had to be stopped by a group of F.A. officials from wrenching the cupboard door from its hinges to reach the hapless referee."
Anyway, it is not Fatty Foulke in the City clip. Therefore, it is unlikely to be Sheffield United.