It's a defendable and logical position to be supportive of excellent state education for all which eventually renders private schools obsolete, and voting and campaigning for that aim, whilst simultaneously back in the world as it is here and now choosing that in the absence of such a utopia one might still spend £15k per year on school fees rather than the skiing holidays, gym memberships and other luxuries that one might fritter the money on, if the local options make that the best choice in the current landscape.
In reality changing state schools to provide adequate education is going to take way more money than can be clawed from the rich via VAT on school fees. Per head funding should be at least £10,000 per year to give schools enough staff, equipment and resources. That would cost an extra £50bn per year, which is an average of £1600 per tax payer, an increase of just over 30% on current levels. It would take some quite drastic culture shifts to make that generally acceptable.
You could do it by converting private schools, and all state schools rated as meeting a new "even better than outstanding" rating that means they are everything we could wish for, into something like the direct-grant schools that were abolished in the 70s whereby there are some state places and some fee-paying places but to get a state place you have to opt in irrevocably to a higher-level income tax structure with rates set such that once the majority have opted in, that extra £50bn is available. Government spending can channel funds to state schools until they all meet the standard of excellence that would be the new standard. (And once less than 50% of the population is still on the old tax system that could be phased out.
No of course it wouldn't work.
Meanwhile because schools don't charge vat on the services they provide, they can't (as other businesses can) reclaim the vat they pay on purchases like computers and other equipment. Charities are not exempt from paying VAT like this whereas businesses effectively are, so simply adding VAT to school fees would raise less than you think.
They would have to invent a new non-VAT tax which applies only to private schools of the kind that they want to target (somehow exempting nurseries, home-education with paid tutors, schools specialising in SEN, schools for ballet and dramatic arts and schools for children of military personnel and all the activities of educational charities that aren't schools) so that normal VAT reclamation didn't apply.
God knows what would happen for schools where some pupils come into an "exempted" category and some don't.