I don't really have an issue with adding VAT to state boarding schools, except that it would be expensive to implement just for 32 schools, because of the mixed VAT regime it would require. It's not a reason not to charge VAT on private schools.
If it does reduce the number of people in private school (and it won't by much) that's another societal benefit of tax.
On the other hand, as almost every single private school child has a parent in the top 5% of earners, they will simply stump up, and it will be a nice earner for the Treasury. It's a beautifully targetted tax, and the potential downside if it doesn't raise tax is not anything to worry about.
I am not going to weep for a bunch of wealthy people paying a bit more tax, or sending their kids to state school if they chose to do so instead, like the other 93% of the population.