Wouldn’t Dec 31st, 0000 be considered the last day of the first year?
It would have been if there had been a year 0, but there was no year 0.
The whole problem is that there is no “year zero” in the calendars. The concept of zero, which was invented by Hindu mathematicians and transmitted to the western world by Islamic mathematicians through the use of Arabic numbers, was not applied to the western (Julian and then Gregorian) calendars. I don’t think it was applied to the Islamic (Hijri) calendar either. Now that I think further, the calendar was developed long before the concept of the zero. This may be a part of the confusion.