Of course not. Gregorian Calendar didn't drift, it just fixed drift in Julian calendar, which was like a couple of weeks by 1582. And "new year" in Julian calendar was 1st of January, same as now. It was inherited from previous Roman tradition, because Romans already had the custom to mark 1st of January as a new year, because since 153 BCE it was the date when consuls were inaugurated. So it's an entirely political thing and makes no real sense whatsoever. We are celebrating the day of Roman consulate inauguration for more than 2000 years now.

And before that they they started years from 1st of March, which is much closer to one of the Equinoxes, which is what all sane people (including french revolutionaries) consider to be the proper start of a year.