Here's my opinion on private schools. Do they have a built in advantage? Possibly. Some private schools in the past have required students to go out for at least one sport, which could create big numbers out for a sport. This gets those borderline kids out for a sport instead of them just going fishing or whatever. When Rockhurst goes to a cross country meet, they bring 70+ kids sometimes while other teams may have a dozen.
2) If a parent can afford to send a player to a private school, they likely can afford to get them private lessons for golf and tennis, send them to the best football camps around the state or beyond, afford to pay entry fees for the best off season basketball leagues, etc.
3) Success builds success. A few years ago, if your kid in Jefferson City was good in youth wrestling and you could afford it, they were sent to Helias which dominated State Wresting. If you're in St Gen. and your kid is good in youth football, and you can afford it, why not send them to Valle with their state championship tradition
Are rules bent? Probably, in some cases but not always. I used to work with someone who attended a private school that told me exactly how it worked at that school. Hear about a youth that's tearing it up on the basketball court in middle school. Find out what their dad does for a living, Have an alumni in that specialty offer Dad a job, viola! There's your next all state point guard. I doubt this is common place, but it likely happens occasionally.
Are rules bent in public schools? Occasionally. I knew someone years ago whose uncle was superintendent of schools. They were on the edge of being bumped up a class so the uncle allegedly with held a few "late enrollments" when he turned in his numbers and the school stayed in a smaller classification.