I know a guy who's learning fiddle. His main problem is bowing - he can't get the hang of moving the bow in a straight line perpendicular to the strings. Personally I think his teacher isn't hammering on that as much as they should be. So to answer that question: the teacher shouldn't care if the student prefers to play a different way, but they *should* be digging their heels right in if the student's technique is incorrect for how they're choosing to play.