So many issues with that assertion. I'll start with a few.
Private school parents already subsidise state schools as they pay for places they don't use. The average saving per year to the tax payer is 7k per child. This hugely outweighs any benefit the private school tax status infers.
The bulk of the tax benefit is used for charitable purposes eg funded places for lower earners. These will all disappear overnight if the tax position changes as the other full fee paying parents understandably won't want to fund them.
If any pupils leave the private system the state will have to pick up the tab. The total raised from VAT may not even cover this meaning the average spend per state pupil will actually decrease.
Even if not a single person left state school and all remained and paid VAT the total amount raised would be around 1% of the state education budget. That would cover inflation for a couple of months and would likely result in not so much as a new pencil in most schools.
If the intention is fairness then this policy fails badly. The elite will be able to pay the VAT without issue and will continue to use the schools. The families paying fees through parents with professional jobs may be able to continue. Those who make huge sacrifices to fund it or are beneficiaries of bursaries will likely be forced back into the state system.
The end result will be an even more elitist private school sector and no discernible change to the state sector funding.
A further interesting postscript will be to see how this plays out in practice. Starmer and Labour who are so keen to align with the EU will have to introduce legislation that is illegal under EU law where VAT on education is prohibited.
He could of course still go ahead but then the question is how do you define the category of service that attracts VAT. General VAT policy states that it can only be levied on widely defined products or services eg you couldn't have VAT imposed on children's shoes but not on children's coats. The entirety of children's clothing either attracts VAT or it doesn't.
Where education is concerned this could mean that if they want to make private schools fees subject to VAT then they will also have to add it to all educational services including nursery fees, university fees, private tutors etc.
As I said it is a nonsense policy from an economic perspective but sounds good to those who don't understand the detailed ramifications of it.