I can only give an anecdotal response to this and some thoughts. Hopefully others will chime in too.
My D used her high school music teacher for recommendations. We did not use her private teacher (as she was not “famous” but she worked at a well-known music school in the city and was known in the city - still that was on my daughter’s music resume along with her name - for most schools they would not have known her).
Her high school teacher had given my daughter guidance on schools. It was a big school and each year a few students went to good music schools. We knew she would write a thoughtful recommendation. Plus she also knew people at the good regional schools like Lawrence.
We also used her high school director for the MT auditions. He had told her he would write a very good recommendation. In one MT audition, they had his recommendation in front of them and asked her to do some “things”, mainly vocally, that he mentioned in the recommendation. It was the only audition where the recommendation was “present”.
Besides knowing that the teachers would write good recommendations, at my D’s high school, all the teachers recommendations were loaded electronically into “their system”. I’m not sure how we would get outside teachers recommendations loaded. Something to think about. My D needed music, acting and academic recommendations. So most recommendations were just loaded at the school and sent electronically to the universities. We never touched them.
However I do remember that the MT recommendation, plus additional material, had to sent in a separate envelop. That was the one where they had definitely read it and considered it. Also another school did require the music recommendation to be sent separate as well - because it went to the wrong address and was returned to the high school - that was fun to track down!
It’s such a busy time and getting applications done is like trying to dock the titanic. So her high school and teachers made it easy. And I never considered asking her private teacher…maybe my bad… We were winging it a bit.
My guess would be if you have a well-known instructor, by all means try to get a recommendations from him/her. Just be sure that whomever writes the recommendation has an interest in your kid and will do a good job.
So consider who will write a very good recommendation, on time!, along with how you will get it there. You may want to ask the private instructor if they do regularly do it and if they would be comfortable. Then go from there. I still think high school teachers are a good resource if you trust them and have a good relationship with them.
Hope this helps you!