"Folksingers don't play it and no folksongs of note are written about it

I think you must be talking about the local game played in the United States and known there by that name.

Football chants at matches in England aren't the greatest songs, but they satisfy most of the criteria for being classed as folksongs.

"Do they sing some sort of patriotic hymn before sporting events in the UK" Leaving aside God Save the Queen at International matches involving England (but not Scotland or Wales), there is "Abide with Me" at the English FA Cup Final. The words aren't exactly patriotic, and all the better (I think it was King George V suggested it, back in 1927):

Abide with me; fast falls the eventide
The darkness deepens; Lord, with me abide
When other helpers fail, and comforts flee,
Help of the helpless, O, abide with me.

(Hell, now the threads down in the BS section, it's only right we should make a point of putting in songs...)