For example, in England we have academies, which are exactly what you described, directly accountable to the Department of Education. However, we also have community schools, which are accountable instead to local authorities, and we also have voluntary aided schools (like faith schools), which receive some state funding through the local authority, but are primarily funded and run by a foundation.
You still get to choose where your child goes to school, and if you really want to you can also homeschool them.
Private schools are not about choice, they're about quality. Unfortunately, that comes at the detriment of the quality of state education, and ultimately social equality and mobility.
The authors argument revolves around removing choice so everyone gets the same thing so we are forced to make that one option higher quality. I'm not saying that is the only way to do it, but that is what this person is arguing for and that is what I'm arguing is a bad idea.
The author's argument revolves around removing the choice of paying for quality, not removing choices of the type of schools and education available.
If we got rid of private schools, we'd still have all the choice, but the choices and quality would be available to everyone, not just rich families.
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.
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.
The problem is I think the author making this specific suggestion on how to achieve that is having a hard time understanding the implicit statement they are making on the other side of this coin: "you don't have the right to decide how your child should be educated, a committee will decide intimate details of yours and your family's lives, where they go every day, what happens when they get there, what the curriculum is, what happens if it doesn't go well, what values should be encouraged, what narratives will be portrayed'.
For reference I'm not some radical libertarian I'm fairly centrist and progressive, but the ability to have autonomy in deciding what is getting blasted into your kids head every week day for the first 17 years of their life is not something to scoff at. If that means not every single person gets the exact same opportunity I'm ok with that (and outside of possibly this person's utopic vision that would be true in the other scenario as well).