'public school' drifted into exonymic status as of the middle 20th century in the U.K..
For a long time, as long as I've been alive, they've called themselves 'independent schools', which in fairness is indeed clearer terminology. They're (more) independent from the government than state(-owned/run) schools.
isc.co.uk/about-isc/associatio…
The legal terminology is also and has long since been 'independent school'. See the Education and Skills Act 2008, for example, or indeed the Education Act 1944.
This was observed in 1985: "in recent years the less emotive phrase 'independent school' has slowly replaced the more familiar but totally misleading term 'public school'."
The humorous fact that 'public school' means something that seems contradictory to non-U.K. people has outlasted the very piece of terminology it is talking about, which in fact largely disappeared in endonymic or formal exonymic use during the times of Margaret Thatcher.