Private schools are legally obliged to charge 20% VAT - they may take steps to reduce costs to lower the overall charge but 20% on top of what they charge is now an obligation.
The govt are preventing some kids from going to these schools due to the cost increases: hence the court cases.
Some private schools are businesses, but many are charities (= zero profit)
But even if you think business (or "luxury" for that matter) is a dirty word - if it's core purpose is "education" then then how they are incorporated is legally irrelevant - See VAT on healthcare
Not sure how you could have brought a case against the Govt for VAT on the grounds of ECHR prior to VAT being in the finance bill - but I admire your intellectual contortions too