I seriously doubt that such a situation is true in more than a very SMALL minority of counties. Saying "many, many, many" says precisely nothing advancing your case. I grew up in a dirt poor farming county (though they call'em parishes in Louisiana). It had a WELL-attended Catholic school, and when "desegregation" hit, several secular private schools were started, most of which are still around, and still viable today.
And unfortunately for your position, that "price-fixed monopoly" is rapidly pricing itself out of the market wrt private schools. I would lay a small wager that even in your "county", a private or religious school, properly run, would succeed.
And, of course, there is always home-schooling, which is evolving rapidly into "network schooling" in collaboration with "internet schooling".
The "price-fixed monopoly" more serious attack today than it ever has been. It is a dinosaur, and soon to die.
Not only is my county completely without a private school but so are the surrounding counties.
I did a search for our state and counties ( especially in rural areas) without private school alternatives predominate.