No, it’s really as simple as in my own reply: the American meaning is exactly what the term conveyed when ‘public schools’ began, centuries before today’s ‘state schools’ existed anywhere.

It was a school that was open, in principle, to anyone. Think ‘free as in speech’ vs ‘free as in beer’, but with the added sense - like ‘public transport’ - of being democratic and round-about-accessible to all.

As for the US: when it began, fee-paying schools were the dominant mode, and there wasn’t really an aristocracy with private tutors to distinguish it from. So it never needed the ‘public’ - and when government schooling became a thing, it pretty naturally took on the ‘private’ qualifier instead.

Actually, I think it was that a subset of private fee-paying schools were set up to prepare students for positions in ‘Public Life’ ie politics, military, clergy, civil service - basically running the country aka “The Ruling Classes”.

Nope. Like I said a moment ago[0], it really did just mean ‘free as in speech’, like not-quite-free public transport suggests. Your explanation is certainly very neat and plausible - all the makings of a folk etymology - but it happens not to be correct.

[0] https://news.ycombinator.com/item?id=31008161

You’re right about “Public, as in open to all (who can pay)” being the primary origin of the name, but it has also evolved to mean:

https://www.merriam-webster.com/dictionary/public%20school

“An endowed secondary boarding school in Great Britain offering a classical curriculum and preparation for the universities or public service.”

The purpose of public schools isn’t academic excellence (though of-course they support it for their most able pupils) but networking and the ‘habits of person’ necessary to compete socially in class-sensitive institutions.

Run by -- and for -- the public not by/for the church.

The state did not run schools when these were established. It's much older than that as a term.