At no time in the history of the early Christian church was it ever thought that the birth of Christ took place on 25 December. In fact, Jesus' birthday was not celebrated at all until the mid-4th century. Oh, and the lighting of candles during Christmas Eve services? Another Roman tradition, celebrating Sol Invictus. The 12th century Syrian bishop, Jacob bar Salibi, records, "It was a custom of the Pagans to celebrate on the same 25 December the birthday of the Sun, at which they kindled lights in token of festivity. In these solemnities and revelries the Christians also took part. Accordingly when the doctors of the Church perceived that the Christians had a leaning to this festival, they took counsel and resolved that the true Nativity should be solemnised on that day."
In other words, all of Christendom knew that Christ had not been born anywhere near 25 December; however, since the Roman holiday of Saturnalia was still very popular, it was much easier to merge the story of Jesus' birth with a celebration that had already existed for well over 800 years. And in actuality, Christmas only usurped Dies Natalis, Sol Invictus, when Church leaders cleverly merged the sun god with Christ, 'The light of the world'. All of this was actually common knowledge, even through the 18th and 19th centuries. It wasn't until the 20th century that the masses started confusing 25 December for Christ's actual birthday. Should this change anything for Christians? I wouldn't think so. One could think of it like how the British celebrate the Queen's official birthday in June, regardless of the actual date of the monarch's birth.
Oh, and that jolly fat man who breaks into people's houses to deliver presents, that certain groups of easily offended people swear is white? He's actually a brown-skinned Turk from Asia-Minor..