at new year / on new year’s eve

Now that has confused me a lot lol
I have never heard or been taught that it is also said “at New Year’s” but “at New Year” (no ‘s). In fact, all the pages on the Internet and books I have found say “at New Year” and “on New Year’s Eve” but nothing else.
My questions are, why AT and ON for these cases? And what’s the difference between them?
You, Jedi5384, say “AT” refers to the few days around that time in general, do you mean the last days of the year? If that is what you mean, then “on New Year’s Eve” only refers to the last few hours of the 31st of December doesn’t it?