Lack of New Year related songs

So the Xmas and New Year holidays are behind us. As usual, there were a lot of Christmas songs and movies in the last month. It's easy to name just a few songs: Jingle Bells, Jingle Bell Rock, Last Christmas, Santa Claus is coming to town... But what struck me this time is that there are almost no songs related exclusively to New Year's eve/day. It is mentioned in some songs, but it rarely plays a major role.
Googling I found out that Auld Lang Syne is traditionally a song sung at New Year's. Then I remembered ABBA's Happy New Year. In Slovenia we also have a song by Alfi Nipič, Silvestrski poljub (New Year's Eve's kiss). I believe there are more hidden somewhere, probably in non-English languages and maybe in cultures, where they celebrate New Year's day and it's not so close to Christmas or some other big festivity.
That's because I think it's Christmas' "fault" New Year doesn't get more songs, as it steals the spotlight. Which made me think some more and I realized, that they are basically the same thing! Both are festivals of light. Christmas replaced the old pagan winter solstice celebration, where they celebrated the arrival of Son's child and the start of the days lengthening again. And for New Year's day we throw so many lights in the sky (fireworks), that it's like we're trying to shoot the night away. I also think that for symbolism's sake having the start of the year during winter is bad, it would be better either at he beginning or end. Sure, on the calendar winter has just started, but it was cold much earlier. What we had in the days of old was, in my opinion, better symbolism (Haloween and carnival dates for New Year's day).