The magic of this world is so intriguing, I love the way that it’s “woven" info the lives of the people using it. Just imagine the incarnation of death shows up for new years and is like, “nah let’s sing a bit and bless this shit out of this place”.
I didn't either, but it makes sense that they use a different song. Japanese words were set to the tune for "Auld Lang Syne" a long time ago (the title is "Light of Fireflies" if you want to look it up) and it is used at educational events like graduations. So it would be confusing to them to use it at New Year's.