Why did the Church change the date of Christ's birth?

[Christopher]

It had to be more than just to eclipse a pagan holiday; were they so insecure in their ability to preach the word and have the people believe it that they had to dangle a carrot? "Here, you can still have a holiday on this day, but now it'll be about Jesus instead of your garbage!" further, they had to know that someday it would come into question, and with enough probing the truth would be unearthed? I don't doubt that Christ existed nor that He was the Son of God, I just doubt the wisdom of the Church in telling these small lies because as some people think, a small lie often covers a bigger one. Why bother raising those kinds of questions?