Not having to pay VAT or corporation tax is a massive discount for many private schools.
I've never understood this argument (my hands are clean, as I have never either attended myself, nor sent my children to, anything other than state schools).
The government makes a massive profit from private education. The parents pay the same income and property tax as everyone else, but they don't consume a school place. Suppose for the sake of argument all private schools disappeared off the face of the earth tomorrow morning. The state would need to provide an additional 8% of places, but would be receiving not a single penny of additional revenue. Indeed, government revenue would probably drop: the largest expense by far in education is salaries, and those salaries deliver income tax and national insurance, and the recipients of those salaries spend money which flows through the economy.
The most likely destination for the money currently spent on private education if it ceased to exist would either be paying down the parents' mortgages, being paid into the parents' pension schemes or being spent on (imported) cars and holidays (abroad): none of these things deliver much in the way of tax, nor does the money flow through the UK economy.
The VAT issue seems very confused. If the purpose is to dis-incentivise people from using private education, you need to explain where the money is going to come from to fund the extra state places. If the purpose is to raise more money for state education, you need to explain why people using private education would all be able to handle a 20% increase in fees. All it would take would be for one private education using parent in about six to decide to go state and you'd end up spending more (on additional state places) than you raise (by VAT on those that stay in private education).
I would like to see private schools failing to meet their charitable obligations be complusorily purchased and turned into free schools.
And the financial effect of that would be negative. Massively negative. Free schools are paid both capitation, on roughly the scale of maintained schools, and a contribution to their capital costs. And they don't charge fees. So the effect of nationalising a private school containing 500 children would be additional cost to the state of about two million per year (the capitation for the pupils), plus the capital cost of acquiring the asset, in exchange for not a penny of additional income. And the parents who are currently paying fees would spend the money on mortgages, pensions and holidays (see above) so the money would be lost to the UK economy, rather than the staff at the school paying tax on it and spending in the UK.
I understand the political objections to private education, although I think the cure of sequestration is rather too Stalinist to be justified by the disease of privilege. But dressing it up as "private education costs the state money" is just bollocks: the parents are paying the same tax, and are paying more on top which goes into the UK economy, while not consuming a school place. That's a massive net profit to the treasury and to the wider economy, however you dress it up.