Well, I'm not sure. Can I offer a different view?
On an average private school fee of £20k, vat would be £4k, which is equivalent to what the state saves by not funding a place for those children in private schools.
Also, as a rule most educational matter is vat exempt. And if charged in one sector would it then be applicable to nurseries, universities and so on?
Finally, IME many teachers - unless drawn by the prestige of just a handful of schools - prefer the state sector, as private schools have very long days, especially when you include all the before and after school provision, and often Saturday school.
There could be a complusory period when the tracher had to work in the state sector or repay costs of training - that was suggested for the doctor drain but didn't go down well.
Change is needed but I don't think this is the answer.