It is quite possible and quite probable that Jesus Christ was NOT born on December 25th as that day was the old Roman Winter Solstice celebration that was taken over by the resurgent Christian church.
However, we have now been celebrating Christmas on December 25th for 2 millenia. So, if and when Jesus’s birth certificate is found with a March or June birthday (and mark my words, we will find Jesus’s BC before we find Bam’s BC!), I will still celebrate December 25th as Christmas.
Theologically I dont think churches would care. The Lord was born and the world was saved and if we take one day to honor Him, it does not theologically or spirtually matter if that was the exact day He was born.
Plus a summer Christmas would not be as much fun. Santa in his heavy winter coats would have a serious case of BO
FYI, it wasn't an "old" Roman celebration, it was the feast of Sol Invictus (the unconquerable sun), which was a late Roman cult--dating from 274 under Aurelian. Christianity predates this cult, and Chrysostom in the 380s was saying that the feast of Dec. 25th was established by the Roman Church from the old Roman census records. It is quite conceivable--even probable--that the Sol Invictus cultists took the date from Christianity and not the other way round.
By the way, the earliest mention of both dates occurs in the same place: the Philocalian calendar or Chronography of 354. The Calendar certainly mentions Christ's birthday on Dec. 25: "VIII kal. Ian. natus Christus in Betleem Iudeae." The reference to Sol Invictus is somewhat more tenuous: "N·INVICTI·CM·XXX". Both of these feasts were evidently well established by 354; there really is no evidence to support the idea that the Christians got this date from pagans.
But I agree with the second part of what you said. Theologically, the subject of the feast is the important thing and not the date.