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.