The only problem with that date is that the shepherds were in the fields with their flocks when our Savior was born. It's far more likely that He was born in the spring because of this.

Not that it matters.

The Catholic church moved dates and changed information as needed to try and make Christianity align with foreign deities so they could say "Look, you're already worshiping what we worship, you just got the names wrong." Enjoy your fertility symbols on the Passover...