As an argument this does not, in fact, work

Apparently VAT on school fees will have no effect:

But Sir Keir told LBC Radio: “We have obviously looked at reports on this and all the reports show that it’s unlikely that parents will take their children out of schools.”

He added: “I have looked at this question of will it lead to children leaving private schools and going to state schools and the answer to that, on all the evidence I’ve seen, is no that it won’t. This is the VAT paid by schools, they don’t have to pass this onto the parents, they can do it in other ways.”

So, the claim is that substantially changing the price of private schooling will lead to no change - at all - in the demand for private schooling. Or, as we can also put this, demand for private schooling is inelastic with reference to price.

Except when we say inelastic we don’t actually mean no change, we mean not much change:

A good's price elasticity of demand is a measure of how sensitive the quantity demanded is to its price. When the price rises, quantity demanded falls for almost any good, but it falls more for some than for others.

If your claim fails even Wikipedia’s explanation of the point then it really is possible that your claim is not true.

That second claim is even more silly. VAT is a tax upon consumers. The only way that it can ever fall on anyone else is if demand for the good changes and so shareholders, the capitalists, end up carrying the cost as consumers desert the item and so the business shrinks. Not that private schools have shareholders to dump it on but that’s another matter. The only way that the VAT is not passed on to parents is if price elasticity is large.

What determines how much prices actually rise and, therefore, how much of the tax burden is borne by consumers rather than retailers? It is the sensitivity of demand to prices, a concept that economists formally refer to as the “price elasticity of demand”.

If demand is very sensitive to prices, known as demand being highly “elastic”, then retailers will be very reluctant to increase prices despite the tax, because demand will decrease by a large amount, and their sales will decline dramatically. Under these circumstances, they will prefer to raise prices by a small amount only, bearing the most of the burden of the tax.

….

In contrast, if demand is highly insensitive to prices, or highly “inelastic”, then retailers will readily increase prices because their sales will decline by only a small amount. That means consumers will bear almost all of the burden of VAT.

If elasticity is low - or actually zero, as claimed - then yes, folk won’t take their kids out of private schools. But if elasticity is low - zero as claimed - then the 20% VAT will be passed along to parents. They’re the same statement.

It is not possible to have a Shroedinger’s Elasticity, where it’s high for one part of the argument - non-passing along of VAT - and also low to zero at the same time, no change in demand.

Of course, our pointing this out isn’t going to make an iota of difference given that the idea is really driven by a hatred of any parent not using the state propagandisation service for their children but that’s another matter.