One of my Romanian sons asked if we used the word “Merry” for anything but “Merry Christmas,” because he didn’t think he saw it anywhere else. I could think of quick examples, of course, but immediately noticed that they were all from rather archaic phrases that have continued into modern English entirely whole. Eat, Drink, and Be Merry. Merry old soul. God Rest Ye Merry. We don’t use the word much now, not to zip into a sentence as a synonym for cheerful. It is nearly obsolete, but its yearly commonness in Merry Christmas keeps it alive. The other phrases survive because they don’t need explanation to be understood. If you grew up with Merry Christmas, then you know what it means that Old King Cole was a merry old soul, even as a youngish child. You might use merry, in fact, to help you get the meaning by context of soul in that rhyme. We don’t use soul that way very much anymore. Poor souls.
I might use it, but I often use slightly archaic or formal words to mildly comic effect. It’s not going to last. God Rest Ye Merry is already invisible enough that most people think that “Merry” goes with “Gentleman” rather than the first three words of the carol. As Happy Holidays and Season’s Greetings push out Merry Christmas, it may be that in two generations the word “merry” is not always understood. It might become one of those words like yule, or nativity, that are kept around only as antiques, a sort of museum-piece. The other phrases with “merry” will then fall out of use altogether, dragging “souls” down with it.