In my district, specials teachers who are part time can study for the praxis (that general education teachers take) and increase their hours by also doing RTI. It depends on the district/state/board policy.
I agree that a part time music teacher can earn more by providing outside music lessons. It’s also very possible that a true musician will not want to do RTI- but you might not get health insurance and your pension will be impacted by the lower salary if you are not full time.
Ideally, this student can work as a full time music teacher with full benefits. She can supplement her income by teaching on the side.