The Church is as much a political body as a religious one. Here are Romans, supposedly enlightened, wanted their people to be enlightened, made RC the official religion of the empire. Think of it as how everyone needs to take Rizal in college since it's the law.
But the Empire was very big. They have pagan people in their lands, so off they went and did some c*ckblocking. They knew some of their people are doing the Saturnalia on December, so they did their own Christian thing at around the same time. Here's a neighbor, who isn't your friend, whose birthday falls on August 1st. You aren't invited, some of your friends/neighbors are going but you don't want them to. So you throw a more lavish party on the same date to give your friends options--though you really want them to pick your party over the other guy's. Every year you keep throwing the party, over time you guys became friends so the party merged. But it his still his birthday, he or his folks would celebrate it even if he's attending your party. He'd still be blowing candles on cupcakes at his own home and party with the gang at your place. And so we ended with a Christmas with both the Nativity and Christmas trees and hollies and some pagans doing pagan traditions outside the main party.
I'm not religious but trust me when I say that the original reason behind Christmas was never to trick people. When the Romans became Christians, as what they've learned, they felt they were obligated to lead their people to the right path. But, I'm not as powerful and convincing as those people you've listed.