It's easier to understand the evidence when you appreciate that most early Christians for the first few centuries were semitic peoples (Jews, Syrians), Greeks, Copts and Berbers in the Levant and North Africa. Even the church in Rome used Greek, not Latin, for its liturgy until the 3rd century. There are multiple historical sources mentioning a March annunciation (i.e. conception) and December/January nativity[1] in the 2nd century, well over a century, if not two centuries, before Christianity was adopted by a significant number of Romans proper.

Alexandria and Antioch were far more influential centers of Christian theology and scholarship, where Jewish (and related?) and Coptic calendars were the point of reference. Moreover, during this time Christian theologians were adamant about not mixing pagan Greek and Roman practices with Christian practices. And let's not forget the persecutions. The Roman calendar and the dates of its festivals were largely irrelevant to them in terms of establishing mythological dates or community festivals. And the nativity just wasn't a very important date--and never important enough to be an origin for other dates--to Christians until much later, long after a rough consensus on dates had been established. Theologically the annunciation (conception of Christ) was vastly more important, second only to easter.

There's significant substantive evidence for these dates being established independent of the Roman festival, based on transparent and consistent rationales. The only real evidence for the date being chosen because of the Roman festival is basically a curious coincidence that stands out as significant from a Western European cultural perspective, rooted in a Roman centric historical narrative, by those living many centuries later. The coincidence feels particularly strong and intuitive in modern times, where Christmas is seemingly synonymous with Christianity. Plus, much of the modern narrative regarding the origins of Christmas arises from attempts by Protestantism to delegitimize Roman Catholicism, without appreciating that during this time the church in Rome wasn't nearly as preeminent as they would become [long] after Constantine. The conception of Catholicism as a religion dictated by clergy in Rome who could successfully scheme to set a date for Christmas comes largely from the Middle Ages, after the Great Schism with Constantinople, when Rome no longer had to justify itself to any of the other early Christian churches. This image was later reinforced and refined by Protestantism literally attempting to paint the Catholic church as a malevolently scheming institution.

Now, the spring dates for the annunciation and death of Christ... you could easily trace it back to pagan origins, but rooted in semitic culture, not Roman or even Greek. The sources spell this out for you; no need to connect the dots.

[1] In some instances the dates were mentioned alongside alternative opinions that were significantly different (some web articles seem to mention only these non-concordant alternatives), but that's beside the point. The point is that the relevant dates were rooted at a time when most Christians couldn't care less about the Roman festival, except in so far as they wanted to avoid any conflation.