Ok, I gotta jump in here. Several reasons I believe that all things being equal Public schools have the advantage over the long term, and consistently win over private, and I coach private.
1. Public schools as a rule have a long term history of constantly fielding teams. Coaches,Superintendents and principals come and go but the school, its traditions and pride are still there.
Private programs, like the tides, come and go. They may play for several years and be successful, but if they don't field a team for a year or two, it is next to impossible to maintain any consistency.
2. Both public and private programs have to deal with problem parents, but they can't shut a public program down. In Private, parents with the right last name or right pull can. Public school coaches have contracts, private school coaches have contacts.
For the most part, public schools will have football no matter the won lost record or who the coach is.
Private programs have no public funding and as a result are at the mercy and whims of a few, unaccountable decision makers and when things become controversial or continuous, funding dries up and possibly the program shuts down, only to try and be revived later.
Bottom line, school boards and superintendents provide a buffer for coaches and football.
There is no similar line of defense for a private program, which is why there are so many cancellations during the season. Parents in a private program get together based on some "reason" and hold their kids out.
They couldn't and wouldn't try the same thing with a public school program. They would be run out of town. So it boils down to longevity, public schools with their long term history and traditions will attract the best players that want to be a part of that tradition and history. With few exceptions, pointed out by several others in this thread, private can't compete with that.