The Roman Christian Church at that time has a very different motive. To eliminate any belief that was not their own. That is a very different thing that borrowing a belief from another culture. This is renaming a tradition and denying that the other tradition and worse eliminating anyone who disagreed.
What tradition are they renaming? It's obviously not Yule or some other Northern European festival because they aren't in Northern Europe yet.
It wasn't Saturnalia, as this was on a different day and was even celebrated alongside Christmas in the same societies on different days.
The first reference to December 25 as the Nativity of Jesus occurs in a section of the Chronography of AD 354 known as the Calendar of Philocalus, which, even by this late date, still identified December 17 as ludi Saturnalia.
December 17 was recognized as the date of the Saturnalia as late as AD 448, when it was notated in the ecclesiastical calendar or laterculus ("list") of Polemius Silvius. But now, deprived of its pagan significance, it is identified only as feriae servorum ("festival of the slaves").
Sol Invictus? that's already been debunked in this thread.
So, which festival did they simply 'steal' and rename?
and worse eliminating anyone who disagreed.
"Anyone who disagreed"? How powerful do you think the Church was?
In the ancient world, they had nowhere near the amount of control and power people seem to think. This is before modern communication and transport technology. Before modern police forces and security services across a massive empire with tens of thousands of villages and towns all falling under the control of different political authorities with highly decentralised power structures.
Regional governors frequently ignored imperial edicts, never mind some church directive.
Paganism died out organically over centuries, it wasn't persecuted out of existence. Yes there were persecutions at times, but they were not qualitatively different from the Pagan Roman Empire persecutions.
Religion costs money. Once it lost imperial favour, temples fell into disrepair, without benefactors and all the aspects of a religion based on praxis gradually declined.
Christianity was promoted and gradually caught on. Aspects of pre-Christian myth found a new home and evolved into new myths, as they had done many times before.