I used to teach some.. and I started out with the student on their horse.. and we would do some things. If all was going well.. we went forward.
OTOH when it was clear things were NOT progressing I would get up on Ol' Dobbin myself. Was I a perfect rider? Nope.. never have been.. but I could tell right away if Ol' Dobbin was working right or not. Sometimes the student was not advanced enough to deal with a problem. If I saw that, and it wasn't a big deal, I would get on and give the horse 15 minutes in Elana's Horse training School.. and then the student would get on his/her horse with an attitude more in tune to what we were doing.
Never thought about refusing to get on a student's horse. Fact is, I often asked to get on. Sometimes I got to spend a few minutes on a really wonderful horse that way! ;)