I think you've misunderstood what's changed with VAT. I don't blame you: the government have deliberately used misleading language.
It's not that private schools must start paying VAT on the things they buy: they always had to do that (unlike state schools who got it reimbursed).
It's that they must now charge parents 20% VAT on fees and hand that directly to the government.
To explain the VAT offsetting (since people do get confused about it), eg:
1.Furniture Company WeBuildStuff buys some wood for £100 + VAT, ie costing WeBuildStuff £120 total. £100 goes to the woodsman. The woodsman must send £20 to the government
2.WeBuildStuff turns the wood into a table. They will sell it for £200+VAT (ie Joe bloggs will pay £240 for the table.
3.WeBuildStuff can offset the £20 VAT they paid on the wood (which the woodsman sent the government) against the £40 they got from Joe Bloggs as VAT. Ie they only send the government £20 VAT. They haven't actually gained anything: they had already given the other £20 to the woodsman to give to the government. The government ends up with £40 (which is 20% of the total cost of the table, paid by Joe Bloggs - the end consumer), the woodsman ends up with £100 and WeBuildStuff ends up with £100.
It's just a clever way of gathering the tax, so that the government get 20% of the final cost in total, without repeating another 20% at each point along the supply chain. Each company is responsible for collecting the VAT on 'the value they have added'. Paid for by the end consumer.
For private schools, they already paid VAT on anything they bought. But because they didn't charge VAT on fees, they weren't able to offset it against any VAT paid by parents. They were counted as the 'end consumer' of those things they bought.
Now that education itself is VATable they collect VAT on fees. The private schools will now effectively not pay VAT separately on the VATable goods they buy (as if they were an end consumer): instead it's rolled into the VAT the parents pay on everything: teaching as well as other VATable and non-VATable costs.
When you think about it that way, it's obvious why the VAT offsetting can't make that much difference. In education the biggest expense is teaching salaries which the school obviously don't pay VAT on (only NI and other employment taxes).
On average private schools costs are estimated to go 20% on VATable expenditure, 50% salaries, and 30% on other non-VATable costs.
Offsetting that 20% of their expenses which are VATable - ie they save 17% of the cost of those VATable goods - means they can reduces fees by 3.5%. ...But now if you add the 1.2% employers NI increase on the 50% they spend on salaries, you have to put fees up by 0.6%. So they can reduce fees by 3%...
So if parents have to pay 20% VAT on 97% fees, their total cost already goes up by 16.5%, even before NMW increases and the newly added business rates (which are harder to estimate).
And that's average - including rich schools with lots of capital spending. Smaller schools will spend a higher proportion of their income on salaries, so you can redo the calculation for them. (A school spending 5% of income on VATable expenses and 70% of income on staff have to spend all the offset VAT on the NI increase, so can't reduce fees at all.)
The government are deliberative misleading the public about this policy. They're using all sorts of divisive rhetoric like 'tax breaks' and 'choosing to pass on the tax' which are blatant misinformation.
Next you'll be telling me that you believe Labour that this will bring in extra money for state schools.. even when the parents who were saving the government £8k per year decide not to...