So, it's education? You did say it isn't education. Or is there a difference between "education" and "an educational service"?
Because the VAT exemption on tutoring, after school clubs, residential childcare, holiday camps, tutoring, music lessons, school trips, drama classes, is part of the same exemption. And to answer my own question, all used extensively by state school families, because those services are part of the private school package as you have rightly said.
So are those things "education" or "an educational service" or "a luxury"? Should they all be taxed? I'm for keeping them exempt, because I want music lessons for state school kids to be available to as many as possible and because I consider them to have social benefit.