Private school would never have been an option for my parents and it will never be an option for our DC. Even if we were loaded, we still wouldn't consider it.
However, for those who do choose it, they've already paid a load of tax towards state school places to which their children are fully entitled, but which they are not occupying. Also, the overwhelming likelihood is that a family who can afford private school will already contain at least one high-earning, higher-rate taxpayer, who is already paying a lot in income tax.
Moreover, considering that a lot of parents scrimp to go private in desperation after they have reason (justified or not) to believe that state schools have failed their children (bullying etc), what a slap in the face it would be for them if the state fails to adequately deliver that for which they've already paid their taxes, thus forcing them to expensive alternative measures, and then seeks to effectively claim a reward for its own failure.
Therefore, I don't think that people should have to pay VAT on private school fees. In fact, there could be an argument (not one to which I would subscribe personally), that they should actually receive tax relief on school fees.
It's not exactly the same thing, but it's a bit like when desperate people, unable to countenance long NHS waiting lists, use their life savings or take out a huge loan to go private - and then are told that, as they have 'decided' to go private, the NHS will no longer treat them.