Let's break this down for you in simple terms, since this comes down to the very basic difference between the two that you clearly fail to understand.
1. Public schools are funded mainly by your property taxes and mainly based on geography. The people that live in a school zone fund and attend those schools. That is a basic definition of a public school. The tax payers in those school districts make the rules on who can attend. However, a PUBLIC SCHOOL DOES NOT SOLICIT STUDENTS, THE STUDENTS THAT ATTEND LIVE IN THE ZONE OR STUDENTS APPLY TO ATTEND WHO LIVE OUT OFTHE ZONE IF THIS ALLOWED BY THE SCHOOL DISTRICT.
2. Private schools are funded by tuition of students and donations from alumni and like-minded foundations. They do not rely on tax dollars. The privte schools make the rules on who can attend. A PRIVATE SCHOOL MUST SOLICIT STUDENTS OR IT DOES NOT EXIST.
Now carry this over to athletics. Private schools solicit students to exist and there is a natural carry over for athletes, who are often OFFERED reduced or free tuition to attend if they play sports and follow the school's rules. Public schools use who they have. That means kids that live in the zone for the most part or kids who apply to attend the school.
Once a private school athletic program becomes strong, top athletes from around the area are either solicited to attend or ask the school to attend, and may end up attending for reduced or free tuition. Top private school athletic programs become well funded and will naturally have competitive advantages over a public school. I realize that this is not the case for every private school, but it is becoming more routine by the year. These schools realized that very successful sports programs create a high profile for the school and create a demand to attend, resulting in larger attendance at higher tuition (except of course for select elite athletes who will attend for reduced or free tuition). In essence, private schools are using an NCAA model while public schools are just that, public schools. Yes, some will attract a lot of transfers, but remember, a private school is essentially made up 100 of "transfers" because those families with kids there chose to attend and pay (for the educational and athletic advantages).
By separating these different species of schools, you create a much more balanced field. I'm not advocating against public schools playing private schools in the regular season or in post-playoff bowls, but they should not be competing in the same field for championships.