How is the "entire country disadvantaged by private schools"?
PPE, History, Theology, Golf Courses, etc seem to be luxuries compared to driving schools which are VATed. Accountancy Training is VATed, Electrical/Plumbing/Carpentry training at private colleges are VATed, Foreign Language Courses at private centres are VATed.
I don't see why universities should have any special position compared other optional higher education institutions.
Attending university is an option. If someone goes to a private trade school to learn plumbing they have to pay VAT - why should history not be VATed when plumbing or lorry driver training is VATed? This is just a middle class privilege paid for by the working class.
Universities should definitely be paying VAT now. Why should graduates have special privileges and be able to avoid VAT when essential trades like lorry driving have to pay VAT on training?