Just a few thoughts on some private schools:
Specialist music and ballet schools (including the Royal Ballet School) have some places which are funded by the government via the Music and Dance (MDS) scheme, in order that exceptionally talented children from lower incomes can access the training needed at that age. There is no state provision whatever for that level of training, and that is why the scheme exists in the first place.
The dance or music teacher spots the child's potential and suggests that they should audition for a place. The parents don't choose to send their kids to that sort of school at all and would normally never consider a private education.
You can't just pick one of those schools and decide to send your dc there, even if you can afford full fees. There is a rigorous audition process, and the school offers places based on talent and potential alone. Funded places are based on parental income on a sliding scale, and are offered to the most exceptional students.
Adding VAT to fees paid by the government MDS scheme would be absurd, and if the scheme doesn't cover the additional VAT, then many parents wouldn't be able to pay either, so they would have to turn the place down.