Yes there are some requirements like "the kids have to be able to read, and they have to take certain core math courses". That's a lot different than having one game in town where everyone gets essentially the same thing.
There are private christian schools that teach creationism and have religious classes. There are Montessori schools which probably have bordering on nothing in common with the christian school above beyond basic literacy and math education. So there is still an enormous degree of flexibility and choices that you have when choosing education in the present system.
I hate to break it to you, but both private schools and public schools are bound to a curriculum set by the district (and typically overseen by the state/province). Short of hiring private tutors for every subject, parents don't have nearly as much control over their child's education as you are suggesting. And even in the private tutor scenario, if they want their kid to go to a decent post-secondary institution, the curriculum will have to be adequately aligned with the standard admissions processes.