December 25th is not the date of the Winter solstice and there are no historical references to devotees of Mithras celebrating on December 25th prior to Christians celebrating the birth of Christ.
You didn't go into much detail on this and I wanted to offer that detail, specifically the "there are no historical references to devotees of Mithras celebrating on December 25th prior to Christians celebrating the birth of Christ."
So, the claim some advance is that the date of Christmas came from Dies Natalis Solis Invicti (Birthday of the Unconquered Sun), which fell on December 25. But there is an issue with this argument, which is that we lack proof that there was any Dies Natalis Solis Invicti on December 25 prior to Christmas being celebrated on that date. Yes, I know people point to Aurelian dedicating a temple to the sun on December 25 of 274, but that's a dedication of a temple, not a declaration of a feast or establishment of a holiday. It is rather difficult for Christians to take the date from Dies Natalis Solis Invicti if Dies Natalis Solis Invicti didn't exist beforehand!
The first mention of Dies Natalis Solis Invicti being celebrated on December 25, as far as I am aware, is from a document called the Chronography of 354, written in... well, 354.
In its calendar it mentions for December 25, ""N·INVICTI·CM·XXX" (apparent reference to Dies Natalis Solis Invicti). This is our first attestation of it. Thus the earliest year we know of it occurring on December 25 is the year 354.
However, we know Christmas was being celebrated at this time, because
in another part of the Chronography of 354 ("ITEM DEPOSITIO MARTIRVM"), when listing commemoration dates, it tells us that on VIII kal. Ian. (8th kalends of January, which is December 25; the Romans had a weird way of stating dates back then) there was "natus Christus in Betleem Iudeae" meaning the birth of Christ in Bethlehem Judea. This is, I believe, the first clear mention of there being a Christmas celebration being on December 25. This means that our first mention of Christmas does not post-date our first mention of Dies Natalis Solis Invicti; it is possible it predates it, in fact, as the "ITEM DEPOSITIO MARTIRVM" is often considered to have been from 336 AD due to the dates of the martyrs mentioned and subsequently put into the Chronography, though this is admittedly speculative as there is no explicit statement of such in it. At any rate, it is obvious that the mention of Christmas is no later than 354.
Now, of course, just because a holiday is first mentioned in a particular document doesn't mean it wasn't being done before; we are missing an awful lot of documents from the past, after all, so sometimes even major events first show up in the history books decades or even centuries after the fact. So it is indeed likely that Dies Natalis Solis Invicti was celebrated prior to 354. However, the same applies to Christmas. Thus we ultimately do not know which came first, and if Dies Natalis Solis Invicti didn't come first, then obviously Christmas couldn't have had its date chosen as a result of it. Even if we must conclude that the date of one influenced the other (which may not have been the case), it is hardly unthinkable that in the context of growing popularity of Christianity, pagans may have wanted to make their
own rival holiday on the same date as Christmas.