This is a common theory but it's almost certainly wrong.

https://www.bartehrman.com/why-is-christmas-on-december-25th...

> However, upon closer examination, this theory encounters several historical and contextual challenges. One of the key issues is the lack of any contemporary evidence from the early Christian period directly linking the choice of December 25th for Christmas to pagan festivals.

> But several decades earlier (c. 203 C.E.), a bishop from Rome, Hippolytus, wrote: “For the first advent of our Lord in the flesh, when he was born in Bethlehem, was December 25th, Wednesday, while Augustus was in his forty-second year, but from Adam, five thousand and five hundred years.” (Comm. on Dan. 23.3.) [...]

> Tertullian, for example, calculated that Jesus was killed on March 25th. If Jesus had also been conceived on March 25th and you count exactly nine months later from that date, you then have Jesus’ birth on December 25th. I think this is the way early Christians came to believe that Jesus’ birth happened on December 25th.

> Moreover, unlike the previous and still most popular theory, this one is mentioned in the early sources!

> A treatise titled On Solstices and Equinoxes, which comes from the 4th century states: “Therefore, our Lord was conceived on the eighth of the kalends of April in March, which is the day of the passion of the Lord and his conception. For on that day, he was conceived on the same he suffered.”

Note that regardless, December 25th was regarded as the date by at least some Christians long before the fourth century. As for the mention of shepherds in the article, we have independent attestations of shepherding in winter in the area, so the question of "why would they be there in winter" is "because that's normal".