My " wild thought " was just that. Yes, a music scholarship. If a student comes to you and says they want to be accepted in a music school, the teacher would be as invested in the student achieving that goal. Hard work, it seems to me, would be expected on both sides.
I know a golf facility where their website boasts about the number of students who have gotten scholarships, reflective of both student and instructor success. Do these students pay more? Likely not, but it is the basis for their fee structure.