There is no "indoctrination game" going on in schools in flyover country except the liberal dominated blue cities in red states. I have 4 grandchildren in schools and they are all getting excellent educations. There is no indoctrination in their schools.
Florida had vouchers. Charter schools sprang up when they were authorized. Most charters failed because they did fare well financially due to mismanagement as amateurs were put in charge of running them. Private schools began popping up in abandon strip malls, operated a few months, sucked up all the taxpayer's money, and then closed, sending the kids right back to public schools with no funding. You cannot tell me it didn't happen because I taught these kids!
Your idea of competition is farcical. I have a great comparison of vouchers versus public schools I wrote years ago that compares full service and convenience store gas stations and how they compare to public and private schools.
Imagine you have those two gas stations. One is required to hire ASE certified mechanics to work on all models of cars. The other hires no one because they don't do that kind of work. The full-service gas station is just that, with attendants to pump your gas check your oil and wipe your windshield. The other provides you napkins and a squeegee! The full-service station has a wrecker to transport your car when it needs to come there not under its own power. The convenience store gas station does not. Full service offers diesel fuel, while the convenience store has no tanks for diesel. The full-service carries tires and basic parts for most model cars, the convenience store gas station stocks air fresheners, oil, and power steering fluid. The full-service station takes all customers, but the convenience store requires you to be selected as a member to receive a discount on their products.
The full-service station is the publics school. They require certified teachers in all subjects. Private schools do not. Public schools are required to accept all students regardless of income, race, creed or color. Private school do not. Public schools transport students to school on buses. Most private schools do not. Public schools are required to take all levels of special education students. Private schools do not. Public schools offer a wide curriculum of elective subjects. Religious private schools require religious instruction in lieu of many electives. Private schools have admission requirements they can use to weed out underperforming students, while public schools do not. The private school is the convenience store gas station.
See the difference and why the costs are different?
