The libertarian argument against public education (pushed by Cato, etc) is that public schooling is worse than private school systems, therefore we should be not spending public money on education, but rather shuffle those resources into vouchers and other ways to fund the private organizations offering private education.

They advocate for starving public education and giving those resources to private companies, which are able to selectively allow students through the guise of tuition. In addition to that, removing government standards allows for private education systems to teach "creationism vs evolutionism" as a "valid" scientific debate. Which is isn't. Same with climate change and all those other partisan talking points. Education shouldn't be wrapped up in a way that can be influenced by funders--do you think that the Kochs and DeVoses are not advocating for "education" that promotes their aspirations and biases? That's called indoctrination.

So yeah. Any libertarian who actively believes that we should get rid of public education in favor of private companies influenced by money needs to rethink their position. The free market isn't the panacea they claim it is.

There's very clearly work that needs to be done within the realm of public education. But giving the chalkboard over to people with deep pockets is a far cry from it.