That's correct.
Profit making businesses supplying educational services (eg the computer coding camp my DC is going on in summer) also don't charge VAT - VAT is generally charged on "luxuries" and things that are the "basic necessities of life" are VAT free (don't get me started on why tampons are a "luxury") - and up till now all kinds of education are seen as "basic"
When VAT is charged, the business can reclaim the VAT they paid to buy whatever items (computers, stationery&office supplies etc) they bought to supply the service, so the school's costs wouldn't go up much if they had to start charging VAT.
There's enough nuance in the VAT system to categorise similar things differently depending on whether specific varieties are "basic" or "luxury" (eg a rich tea biscuit is vat-free but if you double-dip it in chocolate you have to charge VAT) so it would be feasible to say that "basic" educational activities (which would include after-school tuition, music lessons, some kinds of holiday club) and that the more "luxury" end of the education market does attract VAT.
Some private schools aren't providing a luxury or premium service at all, just a good sensible standard of education, sometimes with an additional specialism not available at state schools, and as a genuinely not-for-profit charity, there might be a way to make those VAT free or lower VAT.
I suspect that when the VAT rules are formulated the big "luxury" schools will change their fee structure so that instead of charging £50,000 per year as "school fees" that all incur VAT the fee is suddenly only £20,000 for "school fees" and £30,000 for other (compulsory) services that still qualify as VAT exempt.