DS13 transferred from our public school to a gifted private school a year ago. There were many reasons behind the transfer: very rigid instructional style, very shallow materials, teachers not understanding high-achieving students, teachers who focus on form instead of content, peer issues, general culture, etc.
The private school's academics, it turned out, are far from what we were hoping for. So even though DS is much happier at the current school (good friends, teachers somewhat more understanding, etc), we did wonder what we bought with the high tuition for the private school.
The nice thing about the current school though is that it's willing to be more flexible, and it doesn't have homework. So DS has had the time to do a lot of things that he wants to do, and has the time to do these things very well--including community activities, online academic courses, lots of music activities, regional and national competitions of various sorts. So, since almost all of these activities are DS's own initiative, eventually we figured out: we paid the school so we can homeschool DS on their campus.
I've long had this suspicion but it seems to be true in our case now: that a self-driven and advanced kid might do better in a school that is flexible even with lower academic standards than a school that is very rigid (even with very high academic standards).
Just thought I'd share in case some other families are in similar situations.