In Ohio, the high school athletic association tried to separate the public and private schools a few years ago. However, the Catholic schools initiated a lawsuit and the athletic association backed down. The private schools, primarily Catholic schools, have dominated the state championships over the years because of their ability to recruit players in their region. Cincinnati Moeller at one time had players from 3 states (Ohio, Indiana and Kentucky) back in the Jerry Faust era when the team would have 400 players vying for a final position on the team - this from a SI article back then. The public schools have been at a disadvantage.
Last night my kids’ high school, Olentangy Liberty, and an underdog, dominated Moeller for the Division I state championship (large schools). So on occasion a public school can win.