As I have posted before on threads about private schools, you can't stop people from arranging extra education and tutoring for their children. It's virtually impossible.
I worked in a European country where private tutoring was illegal. You could be reported for doing it.
What happened was that parents got around it in other ways. Children would be sent during the holidays to relatives that were teachers for "childcare." They were sent to summer schools in other countries. Local teachers were invited to "dinners" at people's homes that involved "just having a look over little Johnny's school work."
But the one very obvious issue with the concept was that it highlighted the obvious familial occupation caste system. The children of teachers became teachers. The children of mechanics became mechanics. The children of solicitors became solicitors. The children of doctors became doctors. And so on ... because those children were exposed to the specialist knowledge, understanding and thinking of those professions within their families.
And that's the problem with seeing private education of any kind as a lever of privilege. Children are educated outside the school system everyday purely by existing within their own families.
I look at my own daughter. She stays with my dad every weekend; he's a retired engineer.
She's only five, but already the lessons she's learnt from him are obvious: 'most things can be mended', 'if it's broken, find a schema and take it apart', 'if you don't understand something, get a book on it.'
Personally, having been a teacher in other European countries, I think the way forward is not to make private schools more expensive, but to place outreach conditions on charitable status.
What I'd really like to see is private schools hosting very subsidised holiday schools in maths, science, IT, languages, art, English, history, geography in areas where there's an extracurricular educational desert. It should be normalised for ordinary children in C1 to E families to go to summer schools on various subjects, both academic and technical, run locally.