I wonder if the Medieval sources actually mentioned Christmas specifically ... or if this article's author doesn't believe or wish to acknowledge that the Son of God was incarnate nine months (give or take) before Christmas.
The theological basis that the Messiah would enter into the world in about March is fairly sound, but is perfectly consistent with a Dec 25 Christmas. In fact, Catholics have since the early days of the Church recognized March 25th as the date that Christ was conceived, and therefore fulfilling such prophecies. March 25th, (or, actually April 6 accounting for the Julian calendar) incidentally, is the day that Christ died; the ancient Jews and Christians believed that great prophets died on the day they were conceived. (This feast is called that of the Annunciation, referring to the annunciation of Gabriel to Mary regarding the conception of Christ.)
The historicist arguments are quite presumptuous, ignoring the impact of the “Roman Warm Period,” for instance.
The syncretist arguments, that Christmas has a pagan origin, are thoroughly bankrupt: Some allege, for instance, that the date of Christmas is based on the date of Sol Invuctus, whereas that festival was moved to the date of Christmas in the third century. There existed a Judeo-Christian holiday on Dec. 25th (or the closes Jewish-calendar equivalent) as early as four centuries earlier, when the Macabbeans established that the 25th of Kislev be celebrated as the return of the Spirit of God into the Holy Temple (the Feast of the Dedication).
Niether the Feast of the Dedication, nor the ancient Judeo-Christian supposition that prophets were conceived on the day they die establish for certain that Christ was born on the 25th, but they certainly establish a pious and biblical reason for choosing that day to celebrate his birth.