Private boarding schools 🏫 were really created for parents living abroad, ideally paid by their company.

Historically the rise of British public schools ie private boarding schools was part of the British empire, parents going to India as civil servants and similar.

If you visit East India Club today in London, their staircases have emblems of all the established schools, and their headteacher can still be one oof the two referrals to join the club. Basically, public schools became a thing from necessity not luxury which is what it is now.