You're probably right. There is going to be tension between publics and privates whether there is regulatory involvement or not, split or not. A lot of this tension results from perceptions that the "other side" is trying to take something away from our own side. To many of us that perception is reality but may not be based entirely in fact. In any case, any action or inaction should be grounded by what is best for the individual kid. My son happens to need the extra discipline and guidance of the private school and thrives in it. He has cousins, however, that floundered in that environment and did much better in the public system.
There are, and always will be, a minority on both sides that take actions to bolster their own agendas rather than what is in the best interests of the kids. Those are the ones that end up causing schools to be punished and casting the perception of wrong-doing upon the vast majority of students, coaches, parents, administrators, etc. who are doing nothing wrong. I don't believe a split or regulation of districts or forming of feeder systems will completely eliminate the bad element. Much like the NCAA, much of the policing would have to be left up to the schools and the responsibility for disassociating themselves from those who break the rules would be theirs as well.
An example that would be difficult to police:
Johnny's dad wants him to attend School A because he wants his son to be on a championship football team. Johnny would do well at School B, his home district, and has no other compelling reason (religion, special interest, etc.) for attending School A. Johnny's grandfather lives in School A's district. Dad puts down Grandad's address as Johnny's residence so he can attend School A.
I saw this scenario many years ago between two public schools, but under a district boundary regulation it could just as easily be seen between privates (How do you draw boundaries for T & X?) or between public and private. This would be nearly impossible for an outside agency like KHSAA to detect just from reporting of students' listed residences.