Aside from practical and logical problems with that scenario, the information is simply off the mark. Aggressive co-opting of pagan stuff began in earnest in about the fourth century; much earlier than that, Christians were thinking about the Christmas calendar.
While it's true enough the early Christian authors commented on the providential coinciding dates--that the son of light was born on this day of light's new birth--there is a much more profound theological reason for choosing December 25.
Click here for the extended version of the explanation, by Biblical Archaeology Society folks, in the article, "How December 25 Became Christmas."
Here's the short form:
1. For a long time, Christians didn't much care about the calendar date of Jesus' birth. The infancy narratives in the Gospels aren't actually much help--they're more theological in design and construction than having any intent to provide historical (or chronological) details.
2. By contrast, the date for the death of Jesus was central in early Christian reflection. And the death of Jesus was marked at the Jewish Passover. Soon enough, Easter as a Christian holy day marking the Resurrection included theologically re-interpreting Passover through the lens of Jesus.
*Passover is dated as 14 Nissan
3. By the 200s AD, Christian speculation about the birth date of Jesus was all over the map, including dates in March, April or May. Within a century, tradition had settled on December (although part of the Church world marked it in January).
About a thousand years later you get talk about the pagan associations.
So...how did they land on December 25 if there was never any known specific date, but they didn't choose the pagan festival days?
4. At about the same time--200 AD--at least one Christian scholar did the math:
*Passover is 14th day Nissan
*This is March 25 in the Roman calendar...this is April 6 in the Greek calendar
5. Again, theologically (not "calendrically") the association was made between the date of Jesus' death and the date of his conception....and marking March 25/April 6 as the Feast of the Annunciation, a reference to the date when the angel announced to Mary that she was "with child."
*The date of Conception is March 25 (Roman Calendar) / April 6 (Greek Calendar)
*Add 9 months...and it is December 25 (Western Church) / January 6 (Eastern Church)
To end, I'll quote from the article I linked to, above:
Thus, we have Christians in two parts of the world calculating Jesus’ birth on the basis that his death and conception took place on the same day (March 25 or April 6) and coming up with two close but different results (December 25 and January 6).
Connecting Jesus’ conception and death in this way will certainly seem odd to modern readers, but it reflects ancient and medieval understandings of the whole of salvation being bound up together. One of the most poignant expressions of this belief is found in Christian art. In numerous paintings of the angel’s Annunciation to Mary—the moment of Jesus’ conception—the baby Jesus is shown gliding down from heaven on or with a small cross (see photo above of detail from Master Bertram’s Annunciation scene); a visual reminder that the conception brings the promise of salvation through Jesus’ death.
