I don’t know why this would shake a Christian’s faith though. The date in December was chosen because thats when the Yule pagan holiday was.

You can't see why? It's pretty simple:

(1) Bible and Church preach the infallible word of Yahweh.

(2) The Church has been telling adherents that Jesus was born at Christmas and the bible supports it.

(3) Jesus, if he existed, was most likely born in April-June.

(2)!=(3). Therefore (1) is potentially false, crisis. One would need to accept that the Church is in error, which leads to questioning everything the Church says. Most adherents rely on the Church to tell them that is right.

The solution is to do research and use multiple sources. Unfortunately, there are many bad sources on Christianity out there like Living Waters, etc.