In medieval days in England the only schooling was either the nobility having private tutors, or the church educating its own "workforce".
The public schools which developed (was it in the time of Henry the eighth ?) were open to anyone whose family could pay, hence public schools.
Admittedly this meant the merchants, small landowners and middle classes who could afford it rather than any peasant, so it wasnt the general public which we mean today, and which is the usage in American "public" schools