I don't really understand the whole idea of wanting to segregate private schools from public schools in A/AA. If it were as easy as, "hey, let's recruit some good soccer players so that we can go for state every year," I would understand. However, this is not the case - in fact, it's not allowed under TSSAA rules/regs.
Unfortunately, soccer is an expensive sport in this country. To play for an upper-level club team, a family must have the resources available for travel, training, and many other expenses.
Now, look at your typical (and I stress typical) state hopeful soccer program, whether A/AA or AAA, public or private. A lot of your stronger soccer schools and players come from more affluent areas of the state: West Knoxville (Farragut, Bearden, CAK, Webb, etc.), Franklin, Brentwood, and so on. These are club level soccer players who come together for a few months and play high school.