By most accounts, 25 Dec as a day in the calendar is a historical accident. The boy was probably not born in winter. Calendar problems, ancient Roman holidays, and the proximity to the winter solstice made this a historical game of telephone until a pope just set it in stone (some orthodox churches don’t agree but January is probably only marginally more correct for his birthday).

Traditionally, Christmas lasted so long it usurped solar new years. On the 8th day of Christmas my sweetheart gave to me … a shitload of weird stuff. Mostly birds, for some reason.

Correct me if I’m wrong here but isn’t in UK English “Christmas” still used to describe the whole year end period encompassing the year change. To me they are two close but still separate events with a bit of decorational overlap. So I understand your question why there aren’t more New Years songs. But the answer may simply be: history and tradition. People tolerate Christmas tunes until the 31st and then they’re all cheered out. And NY for most is just a reminder that it’s back to work now.