Singing is part of the Christmas holiday. Hark, what did those herald angels do? Some songs even mention the act, in a fun ironic meta kinda way. No, they are not easy to come across, and some lines will be blurred in order to celebrate thusly.
Go with it.
Riders in the Sky have some virtuosic levity with the connectiveness of all carols with “The Last Christmas Medley You’ll Ever Need to Hear.” It’s largely instrumental, but they’re the cowboys who can swing it. JJ Dion has a “2.0 version” (now with a second verse) that really sings. Schticks to your funny bones.
Wendell Ferguson goes more melodic with “Why Does Every Christmas Song Have So Many Chords?” Slow country swing, deceptively so.