The VAT change is unlikely to do anything to the higher end of private schools. The bottom end is a bit of an unknown. One interviewed said it would but now pupils are back at school there may be more actual info available.
They can now recover VAT on any that they pay due to running costs.
It probably does, but the answer is to improve state education. One Eton kid in an inner city school of 1000 isn't going to change social mobility, one inner city kid at Eton might.