A Christmas tautology

The version of the Christmas carol ‘God rest ye merry, gentlemen’ that I know begins thus:

God rest ye merry, gentlemen,
Let nothing you dismay;
Remember that our Saviour
Was born on Christmas Day.

Wikipedia lists this as an ‘alternate version’ and seems to believe that the proper version goes something like ‘upon this day’ rather than ‘on Christmas Day’, which obviates the problem I am about to outline. But if we do not accept this emendation, we are faced with a somewhat incredible tautology: of course the day on which Jesus was born would have to be Christmas Day, given that it was the day on which Jesus was born! Regardless of the date—the end of December or the middle of July or whenever—the birthday of Jesus would have to, of necessity, be Christmas. Therefore the exhortation and justification provided don’t really make much sense, do they, given that the date is completely arbitrary and only of necessity is Christmas Day?

I’m certain I’m not the first person ever to have realised this. I don’t know why this is making me so excited. Maybe it’s the snow.