Why would the Shepherds be out at night in the winter during Jesus' birth?

Consider the literal meaning of the names of the months September (7th) , October (8th), November (9th) and December (10th). Assume that the addition of January and February to the Roman calendar did not find recognition till the late 1st century, AD. Also, forget the theory that 25th December was borrowed from Roman culture which celebrated the festival of Invincible Sun on that Day. So, the faithful once celebrated Christmas in the 10th month of the year (then December), which along with its name, subsequently got pushed to the 12th place !

Incidentally, as Jesus was undergoing the trial, Peter was sitting by the fire in the courtyard warming himself (Lk 22:55). Did that happen in the hot month of March/ April ? No way. Why would Peter risk getting caught , by sitting near the fire noticed by all ? Yes, it was cold, very cold.

Now, push both Christmas and Good Friday two months backwards. There you have the shepherds tending the flock in open in October and Peter sitting by the fire in January.

This link, which I pasted from @GratefulDisciple here, (which incidentally is part of an answer to another of your questions) asserts that either a September or late December date are "equally plausible" according to the historical and biblical data that we have been given.

In the linked article, in the section outlining the defense of the December date, it is explained that ancient Jewish writings attest to shepherding in late December, as the weather was mild enough.

Temple sacrifices were conducted all year round which suggests that the shepherds would be out at night regardless of the time of year or the weather.

Historical evidence shows that unblemished lambs for sacrifice in the Jerusalem Temple were kept in the fields near Bethlehem during the winter months. The town of Bethlehem is situated about five miles southwest of Jerusalem in the hill country of Judah. The climate is mild, and rainfall is plentiful. Fertile fields, orchards, and vineyards surround the city.

The New International Version Study Bible points out that “the flocks reserved for temple sacrifice were kept in the fields near Bethlehem throughout the year”.

Luke’s gospel describes how angels bore the good news of the birth of the new born king to the shepherds in the fields. After the announcement, they left their flocks to worship him (Luke 2:8-16).

Joseph and Mary probably stayed in Bethlehem for the 40 days necessary to complete Mary’s purification. From Bethlehem, they could easily make the five-mile trip to Jerusalem for the sacrifice for Mary’s purification (Luke 2:22).

There is also a theory that the shelter in which Jesus was born was a place in the northern part of Bethlehem called Migdol Eder. This was a watchtower with a place underneath that shepherds used during the lambing season to shelter the newborn lambs that would later be used as sacrifices in the Jerusalem temple. The prophet Micah, who foretold Bethlehem as the place of the Messiah’s birth, also mentions Migdol Eder: “As for you, watchtower of the flock [Hebrew, Migdol Eder], stronghold of Daughter Zion, the former dominion will be restored to you; kingship will come to Daughter Jerusalem” (Micah 4:8). This theory is used to explain why, when the heralding angels gave the sign that the baby would be “wrapped in cloths and lying in a manger,” the shepherds seemed to know exactly where to look. And it would be apropos for the Messiah to be born in the same place where the sacrificial lambs were born. https://www.gotquestions.org/where-was-Jesus-born.html

Whether Jesus was born in December or conceived in December and born towards the end of September or born in the spring matters not. We are not told to celebrate his birth but to remember his death and resurrection until he comes again.