Christmas is not pagan?

I'm already familiar with the argument that Christmas is pagan but recently, I stumbled on the video below that argued that the date for Christmas isn't pagan (the festivities are obviously pagan, however). How do y'all argue against that?

https://www.youtube.com/watch?v=TE8lVXOsOAc

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The date for Christmas was chosen specifically by the Church to land on top of the celebrations of the birth of Mithras/Sol Invictus in Rome. Mithras was also born of a virgin birth and there was a major celebration throughout the Roman Empire that was held on 25 December. Since the event was so popular the Church felt that giving the newly converted something else important to celebrate would prevent back-sliding.

As for the date of Jesus' birth, the New Testament tells us that shepherds were in the fields tending their flocks. The reason for this would have been lambing season when it was necessary to mind the sheep 24 hours a day. Lambing season would have fallen in late January, early February...so much closer to Imbolc than to Yule.