We often schottishe to a hornpipe. More than being 4/4 it is usually the right speed. Or, given the right circumstances, we do a bouncy polka to them. Often you see schottishes to Andro/Hanter dro (I never know which is called which) tunes.
On the 3/4 front we mazurka to a waltz, even a Mamou Waltz.
As I said, a tune for dancing fits the human physiology or the band aren't dancers - they are session/concert musos. And I have danced to a few of them in my time.
It is a maxim we try to follow when choosing bands for Stroud Ceilidhs "Good dancers make good dance music".
As for Mr Coverley - I can't recall dancing to that tune except for Old Roger. And more - I don't recall that rhythm for other dances - but maybe I don't think. My feet do!
5 time & 8 time waltzes - scramble dancers' brains. Masochists revel in 11 and 17. I don't go to those kind of events!

Outside of dance there is little constraint on rhythm, other than to give others a chance to play it.