My impression is that many composers and performers license through BMI, ASCAP or SESAC. If they do so, there may be standard policies and contracts for how licensing works.
But some don't. If they have their own licenses and contracts, maybe it can get pretty complicated.
I have friends who have a business running a studio that does only re-recording of popular songs so that organisations can avoid paying large mechanical copyrights.
They have a pool of musicians they bring in on a song-by-song basis depending on who they think should be on the song.
These are top professional quality sound-alikes.
They do it for gyms mainly, so they often alter tempo as well, for the gym requirements.
Probably would suit figure skating routines pretty well.
This is partly a vocal piece. Your friends' band would have to include someone who sounded similar to get the effect you suggest.
But I think it is important that figure skaters and coaches try to maintain a good reputation. Doing something that feels sleazy like you suggest might not do that.
On top of that, this was a very customized arrangement of the song, quite a bit different from other arrangements of the folk song, which gives it its unique sound. The arrangement probably carries its own copyright, so other musicians could not copy it without paying for the license - which might well never be given.
Of course, the scandal associated with this issue doesn't give a good reputation either.
I would guess none of us know all the details. We don't know if the Young Heathens or their representative did something strange - like insert a gotcha into the contract, or tried to charge a lot more once they established the program or got to the Olympics. Or perhaps they behaved fairly, and the coach and skaters or networks or ISU are at fault. Or perhaps there was just a misunderstanding.
So I don't know who if anyone is at fault, and wouldn't want to speculate.
Of course, many coaches and skaters and choreographers will probably conclude using music from the Young Heathens is dangerous and trouble prone - and won't use it. I don't know if that is fair to the musicians or not.
As to your suggestion that a skater, etc. might commission an original piece, there are many potential problems with that. It can be hard to predict what you will get, and it might not serve the need - or the skater might simply not like the piece.