It depends on what the songs are that would make them subject to copyright. A lot of classic Christmas carols and songs aren't really under copyright, and so wouldn't be affected.
But SDC, off the top of my head, used over the last several years:
1. You Got a Friend in Me
2. POTC theme music
3. Somewhere in My Memory from Home Alone (Christmas Carol)
4. Money Makes the World Go Round (It's a Wonderful Life)
Probably some others I'm not remembering too, but point is, they got caught/dinged/whatever and that caused them to reassess.
Perhaps the other shows in Branson were properly paying to license songs, or the songs just weren't under copyright. A lot of Christmas staples aren't. But you can't just, say, use songs from Disney movies, not pay for it, and expect that it won't catch up with you. That's entirely on them.
If they wanted to do Santa Claus is Coming to Town, they would likely need to license all of that separately. Additionally, the fact that they are licensed to use the Rudolph characters in a certain context does not likely mean they can do whatever they want with the Rudolph story. That would likely be separate. Chances are they have permission to use the characters in a few clearly specified instances and nothing more.