[Errant]

I tend to suspect that the Christians in the Roman empire were not so ill-informed, stupid, and dull as to get the date wrong after 200 years or so

They were so stupid as to not even be able to get the months of the year right:

September (Septi-) = 7, not the 9th month!
October (Octo-) = 8, not the 10th month!
November (Novem- = 9, not the 11th month!
December (Decem-, Dec-) = 10, no the 12th month

But one small example of the preponderance of the evidence to the contrary.

[Cicero]

That’s because the new year didn’t begin in January back then, but in early spring. There have been various systems, and the calendar has been adjusted several times. Our current system, the Gregorian calendar, came into use in 1582.