Hands up, I personally benefit from the current policy as my kids are at private schools.
But I also think there's a pragmatic view about what a political intervention of this nature would actually achieve. A tick in the no-one has an unfair advantage column? It may be satisfying for some but there's a far wider remit of advantage in a capitalist society, and I don't think it would achieve it in reality.
State schools would have to absorb extra pupils. It's not just a straight £x per head calculation as that implies the current infrastructure could absorb the increase in pupils.
I can only speak for my local state schools but they're already at, or arguably exceeding, capacity so they'd need to build new schools. Not only is that costly but, where I live at least (near London), it's not easy to find sufficiently large plots of land. They're struggling to find plots just to build a few houses. I suspect that makes the net cost would be far higher than the assumed VAT revenue gain > net cost of the extra pupils to the state.
I believe (though correct me if I'm wrong) that private school teachers tend to be paid more than in the state school. Given school budgets are squeezed, that means a drop in salary for those teachers, and a drop in income tax revenue. Some may decide not to stay in teaching, which wouldn't help the shortage in teachers.
The friends I know with kids in state schools pay tens, if not hundreds, of thousands more to live near the outstanding state schools. They also pay for extra tuition. Their privilege buys them an advantage, it's still not a level playing field. I suspect this situation would be exacerbated further if you add in some ex private school parents also wanting to be near the best schools, but perhaps with deeper pockets to achieve it.
I honestly think it's not particularly well thought through, as tempting as the proposal may sound to those who can't afford private schools.