When I was a kid growing up in Detroit and Boston, there were multiple Christmas celebrations in public school. Everyone participated in the celebrations, including the Jews and the many individuals of Middle Eastern descent.
Christmas carols were sung, presents were exchanged and food was consumed in a partylike atmosphere. There were very few people stoking the fires of sensitivity or trying to make everyone feel different. There’s nothing wrong with diversity, but there is also nothing wrong with unity and enjoying overarching fun and good will. When someone said “Merry Christmas,” it was regarded as a salutation of good will toward everyone, whether they believed in Christ as their savior or not.
By the same token, if someone wished you a happy Hanukkah or a Muslim greeting, and you did not belong to that particular group, certainly no one saw a reason for offense or for ascribing hateful motives to the greeter. I believe most people say “Merry Christmas” because they want to invite others to partake of their joy.