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.