You are missing the point, while yes it is about the recruitment of players, it is more about the differences in the Rules for Private versus Public once a kid decides to change schools.
For example Pierce City could recruit Monett players all day, but those players families would have to actually move to that district, and not just one parent both parents the whole family has to move or the box gets checked for athletic reasons, it would and it is expensive to uproot your whole family, selling your home in one school district and purchasing in another shoot it could take a year to sell your home. However for example if Pierce City was a private school those same recruits don't have to move, they really don't have to do anything except change schools, basically check the box they felt unsafe at their public school an poof you are done and in a new school ready to contribute athletically. So while it is a problem that schools recruit, the bigger problem is how easy it is for those kids to move from school to school that makes the difference and those rule differences.
And we are not even talking about a small class 2 private school in a big city recruiting from over 500,000 people versus a rural public school that in their small county might only have 15,000 total people, that is a whole other issue in itself.