Most of the academic day private schools do not have shareholders. About 7% of their receipts go back out on teachers' salaries. They may well save some funds for later projects or for their later bills of other kinds but there is no one taking profits out. This is in every school where any of my children have ever been including North London Collegiate, Haberdashers, Merchant Taylors - day private schools with no shareholders.
However subject to the law the state can do as it likes and market forces will probably then prevail as to where parents send children.
Education used per se since about the 1300s in the UK (and of course across Europe) to be charitable even if the people you taught were rich or opera goers. Labour - Blair I think - changed charity law in about 2008 such that opera houses and schools could not rely on that any more - that education on its own was not the moral good, the charity. Instead you had to help the poor too which is why richer schools had to bring in more bursaries and not very well off private schools had to start doing more for the local community eg access to their grounds
The new measure on VAT is likely to mean schools have less money so they would be allowed to cut back on bursaries and have their public benefit as something much lesser as they are allowed to cut their cloth according to their resources.
the VAT imposition is pretty nasty - it tends to be parents who are reasonably well off but not in any sense millionaires eg doctors and teachers etc who choose to pay for school fees perhaps rather than something else they might buy like a new car or bigger house. They are putting their child first over their own drug or handbag habit or whatever. Whilst I appreciate the argument that 20% of parents at sixth form level who pay school fees are lucky to have the money to fund that and the other 80% may not have the money or politically feel they want to support the local sink comprehensive by throwing their child into it for the good of the socialist cause, it does seem to be a very nasty tax to impose on what is often not particularly a luxury expense. Eg in NE England where I am from grammar schools were abolished in about 1970. My parents who had been to state grammars and became a teacher and a doctor (NHS consultant) paid private school fees for us. There was then and now no grammar school option. It is completely different in richer areas of the country like London and Cheshire where choice abounds. if state school were the same everywhere in the whole UK, Edinburgh, N Ireland, Newcastle, Kent with the same identical schools in the state system fine but that is not so.