Just guessing here, but Christmas has cemented itself in the wider cultural consciousness thanks in part to commercially exploitable traditions like gift-giving. It’s is often celebrated in some way by people who are not even Christian (Japan, for example). There are very specific themes, religious and secular, associated with Christmas that you can write songs about, The birth of Jesus, obviously, but also family gatherings, winter weather (Northern hemisphere bias), the whole Santa mythos and so on.

What Americans associate with Christmas is actually three separate feasts scrunched into one. Saint Nicholas day (December 6th) which is why Saint Nicholas is associated with Christmas, and Epiphany (January 6th) again for the magi, are the traditional gift-giving holidays, but got engulfed by the unstoppable Yule Tide. So you have these three different occasions with their own songs combined together.

New years doesn’t have the same cultural presence.