First, let's talk about the problems with government schools:
1. It trains children to obey government commands from an early age. This is a major factor in generating a population that will sheepishly allow the Republic to be destroyed.
2. It attempts to force the entire range of human abilities and interests through a single education model. This Procrustian system forces the brightest to essentially educate themselves while the slowest take all the resources and the middle gets the scraps. It reaches its logical conclusion in the drugging and/or ostracizing of children that do not fit the model.
3. It is staggeringly expensive. Being a government operation, it sags with inefficiency, redundancy, bloated administration and bad employees. There are little if any incentives for good teaching and almost no disincentives for bad teaching. Because there is no choice, the "customers" are stuck with whatever teacher they get. As a parent, you are essentially forced by law to turn your child over to some petty tyrant or moron. Or you can put them in a private school and pay twice.
4. A handful of disruptive students are allowed to divert the majority of teacher attention.
5. They inculcate a homogenized, sanitized, government-approved version of history.
6. It is funded by forcing one group of people to pay ( a LOT) for the education of a different group of people.
There are many more problems, but those are the big ones.
Public schools have one positive attribute - they are available to everyone. Of course there ARE some really good teachers in public schools, but that happens in SPITE of the system not because of it.
Free market schools, on the other hand, would suffer from none of the problems above - at least not for long, because very few people would be willing to pay for that kind of "service" and would take their money and children elsewhere.
The problems with free market schools are as follows:
1. Although the cost would certainly go down as a result of competition and the absence of government subsidy, not everyone would be able to afford to hire other people to educate their children. But if that bothers you, you could use your own money to pay for the education of others. In that way society, in the aggregate, will get EXACTLY the amount of education it wants (as measured by what each person is willing to pay for).
2. Some schools will teach things you don't agree with. Deal with it. That's what freedom looks like.
To me liberty wins this contest easily.