Having worked and raised a family there is no way sufficient attention can be given running an educational program without something suffering. Most parents are not capable of running a top flight education program having neither the experience nor education necessary to do it.
Public education came about because reliance upon only private educations failed and was generally only available to the rich. There is no way it could have done the job of integrating millions of new Americans into our society as well as public education did during the 19th and 20th centuries. What passed as education prior to the spread of public education was, in general, pretty weak stuff certainly not acceptable for a world like this. Most people of that day could read and write a little but only the wealthy could have been considered educated in any real sense.
Given that locals control the public school systems why would you have any faith that corporations, or churches would do any better? Given that religious schools pay much less for teachers why would you believe the quality of teachers would improve?
There is a lot of propaganda afloat as regards the history of education in American AND about its current state.
My parents were blue-collar, working class people who sent their children to parochial school. My father worked two jobs and went to school at night. Eventually, he got the job he wanted, but then my mother went to work, too. Although they struggled, they were able to afford tuition back then.
Back then, tuition was far lower. But now the public school monopoly has put many Catholic schools in my area out of business. People look at their property tax bills and figure, Well, I'm paying for public school, might as well send my kids there. These days, not enough students enroll in parochial schools to justify keeping those schools open.
Now, my husband and I are raising a family. We worked split shifts for years when our kids were younger. We managed to do it all without daycare or school. Yes, something has to give, and it's usually housework ;-) (speaking of which, I have laundry to do).
But this is not about one type of educational system being superior to another. It's also not about teachers. (There are many teachers in my family). It's really about protecting the right of parents to decide which type of educational system suits their own children. It's about providing people with more options in education, which the private market can do.