I like the idea (coincidentally I already have all of our songs written down, plus a few new ones Jono & I have been musing over) & can get the list up pretty quick.
As for recording, you could do it at a game as long as you had something decent to record sound with. Good luck getting a group of us to come around & sing songs away from the footy, we aren't a choir (though could probably be bribed).
Not a fan of the song sheets TBH, better fan groups have tried & failed. If people really want to join in then they'll pick it up & do just that. They can check the website for any lines they aren't sure of. Song sheets just seems so fake.