The first carols were not songs, nor were they specific to Yule.
For pagan religion is preeminently danced religion.
“Carol”* originally meant a round dance (one of the first recorded uses of the word in English—from 1330—referred to a “carol of the stones,” i. e. a stone circle**), and specifically a ring-dance performed to sung rather than instrumental accompaniment. (They say that when it wasn't safe to have musical instruments at the sabbat, we danced there to songs and mouth-music instead. At the sabbat, you can't not dance.)