I taught renaissance dance for over a decade - and always found people to be a lot more responsive to trying to perform after seeing one done. Seems like it'd carry over to vocals:
Get a cute little kid (nobody refuses cute little kids at Christmas) to give everybody a songsheet - hey, add a candy cane or maybe a santa button - then the M.C./perpetrator gets a pre-arranged group of 2-3 to (enthusiastically) announce "we'd like to sing you a new song-PDQ/old favorite and then you can help us with a few more and get into practice for the holidays" and swing right into it.
PS I agree with you on Frosty/housetop/etc. BORING - and no harmony either, darn it. Elaine the alto.