Exactly.
And I lean towards a June birth.
According to this article, late spring, April/May, is a time of "lambing" (the birth of lambs), called "late lambing", coinciding with the "natural breeding and lambing season". It is after this birth, and the lambs being grown enough, that ewes and lambs need to be pasture fed. June fits beautifully with the shepherds being out in the fields tending to flocks with ewes and their recently born lambs.
http://www.sheep101.info/201/lambingsystem.html
Christ's birth in June coincided with the Feast of WeeksShavuot. During that feast, the Jews remember their deliverance from Egypt, as well as their entrance into the Promised Landthe land of milk and honey. Milk, cheese and honey is the traditional food served at this feast.
The Shavuot also celebrated the giving of the Torah, the Law, on the Mount. Christ was "born of a woman, under the Law". As well, the event of giving the Law is said to be the foundation of Judaism. Christ is the Foundation of His Congregation. Christ born on the feast of Shavuot fits into that.
The eating of milk and honey is prophesied by Isaiah in the very same chapter mentioning Christ's birth by a virgin:
14 Therefore the Lord Himself will give you a sign: Behold, the virgin shall conceive and bear a Son, and shall call His name Immanuel.
15 Curds and honey He shall eat, that He may know to refuse the evil and choose the good.
16 For before the Child shall know to refuse the evil and choose the good, the land that you dread will be forsaken by both her kings.
Additionally, the traditional reading of Ruth is done for the Feast of Weeks. As we may recall, Ruth is about the time of barley harvest taking place during the feast of weeks, as well as giving the lineage of David, and Christ is the Son of David.
So, in Shavuot, with the milk and honey, the remembrance of their deliverance from Egypt, Christ was born the Deliverer of His people, born in the feast of weeks, the Son of David, born of Ruth.
It is interesting to note Christ fulfilled the feasts in reverse: first, the Feast of Tabernacles–He was conceived; next, the Feast of Weeks, He was born; last, the Passover, He gave up His flesh for the sins of His people.