The success of private schools in high school sports can be attributed to a number of things. I'm sure I have missed some, but a few are listed below:

1. They can have a bigger pool to draw from.
2. It is easier to recruit. Even if it's only the parents.
3. It is easier for a kid to transfer if they are not happy with their school
4. People who are willing to spend the money to send their kid to a private school may be a competitive over-achiever, raising an over-achiever kid.
5. Expectations. Private schools have a history of expecting more from the kids in all areas.
6. Parental and community support. People who are willing to pay to send their kids to private schools, tend to get nvolved.
7. Private schools do not have to accept kids who are problems
8. Participation rates in private schools tend to be higher.
9. Many of the parents who have the money to send their kids to private schools are also paying for coaching and other help. This is particularly true in Soccer. It has become an affluent sport with paid coaches most of the year.
10. Tradition builds interest. Rockhurst, like Hutch, has had success. That success will spark interest in kids who would go there anyway, but also kids who may not have gone there otherwise.

Considering that there are a number of bigger private schools that are Catholic schools, there are some unique things the Catholics have or do.
1. Catholic youth sports have been big for a long time. Particularly Football, Basketball and Volleyball. The Catholics have leagues for kids in these sports starting in 3rd grade. The public schools start in middle school, junior high or high school. In many cases, the Catholics have been playing longer and together longer.
2. Tradition. Sports are big in Catholic communities. A Catholic High school without a state championship in football is considered a failure. Someone told me that Aquinas had 40+ seniors on the football team a couple of years ago when they lost to Hutch in the state championship. There is no way they all got to play.

All of these things CAN also be seen in public schools. Goddard and Emporia have strong wrestling programs for many of the same reasons. When Glen Wier(?) was coaching football in Olathe, kids were moving into the district to play for him. It does happen. It has been reported that in affluent areas, parents are offered jobs and the people doing the recruiting explain to the parent what the best schools is, ie moving their kid into that district. In general, you could argue a similar advantage for affluent school districts over less affluent school districts.

Missouri does have a rule that bumps the private schools up a class. This was primarily driven by the success Rockhurst has had in football. While Rockhurst is an example of a school that probably should be bumped up a class and can compete in that class, there are other private schools in Missouri that are negatively impacted and some sports, even at Rockhurst, that are negatively impacted. Rockhurst has NEVER won a wrestling state championship. I don't think they have ever won a regional. I'm not sure if they have ever won a tournament.

I don't think the smaller private schools have the same success or the "advantage". Last I heard, Christ Prep in Johnson County was playing 8 man football. They must not be doing too much recruiting if they can't field an 11 man team. Bishop Ward in Johnson County has had a strong baseball program, but I don't thin they have excelled in any other sports.

My point here is that not all private schools have an advantage and the reasons for the "advantage" are varied. People need to think those things through before "solving the problem".