Not likely, since shepherds were in the fields tending the sheep. It is more probable He was born in late summer or early fall. Some believe Jesus was born during the Feast of Tabernacles, or Sukkot.
Yeah, you can look for clues from scripture about the time of year. Given that there was the time of the census around the birth of Jesus. Also like you say.. the shepherds tending sheep.
People get caught up in doing all the 'religious observerances' sometimes. They go thru the motions without the special focus on Jesus.
Eg.. Having Lords Supper EVERY sunday.. set.. ritualised. Takes the meaning out of it.
Doing Lent.. Advent.. etc.. because thats what it says what time of year it is in the church newsletter.
Christmas can be like this.. going to a service on December 25.. thru the motions.. nothing special about it.
Nothing really wrong with marking the times.. but religious actions and rituals take away from the special meaning. Especially with the Lord's Supper.. given that it is in a church family, and the Holy Spirit will be 'in the midst' of that church family as they take part in it.