One could argue that Tradition is not unanimous on the date of Christmas. If I recall correctly, celebrating on December 25 was the custom in the West and Egypt, while in the rest of the East it was celebrated on the same day as Epiphany, i.e. January 6. The Armenians still celebrate Christmas and Epiphany together on that date. But where did the December 25 custom come from, and why did Constantinople and Antioch eventually agree to it, if they didn't believe it was authentic?
I agree with Joanna that one shouldn't get hung up on these details. I think it's possible to give Tradition the benefit of the doubt, without falling into despondency if the historical facts turn out to point the other way. Either Tradition made December 25 the day of celebration since that was the actual date on the Julian calendar when Christ was born, or else Tradition settled on that day for other reasons. In either case, we can be sure it was the Holy Spirit that made the decision and we have nothing to worry about.
