Robin Hanson asks: "What are the plausible market failures in education that might justify governments running schools, as opposed to perhaps subsidizing or regulating schooling?"
My off-the-cuff answer was Schelling-ish: there's a high enough probability of a separating equilibrium that clusters poverty and corruption together to justify government provision. Of course, this is a terribly weak answer, since the same forces that would corrupt private provision of education will also tend to corrupt public provision.
I think I'd add to the argument a couple of elements: education is important in a few impressionistic senses, and it's important to a median voter whose information set has changed drastically since prior to the rise of universal mandatory schooling. I think it's not entirely unfair to claim that most people credit a combination of labor regulation and universal education with the end of child labor. This suggests to me that (voting) folks will tend to be naturally sympathetic to the notion that the locus of praise center on the valiant political elite. Universal mandatory education was a victory against the non-euvoluntary exploitation of juvenile labor. To suggest the voting public cede control over education to private interests would be to repudiate this important, civilizing victory. With this in mind, in practical terms, the burden of proof to show that governments ought provide education is but goosedown.
Consider a case where 90% of the country is pretty well fed, and can provide adequate if not excellent education for its kids under any ol' regime. The remaining 10% is not so well off, and they might be tempted to keep their teenagers home to watch the toddlers (for example) while the mom is off working a double shift just to cover rent and diapers for the month. The well-meaning majority, having recently watched an investigative journalist's report on such a case sees an easy path to make things better. Here's the problem: mom is being exploited by cruel, profit-seeking employers who pay a pittance, and it's the kids who suffer by being trapped in a cycle of intergenerational poverty. The solution? Easy. Raise the minimum wage and make the kids go to school.
Now, let's ignore the unintended consequences for a moment and consider why the 10% folks might be more likely to see government provision rather than transfer payments. My best guess for this is that the distant 90%, seeing the "exploitation" that already occurs will be loath to subsidize yet another private firm that will prey on the poor.
Now then. In the US, we've got something like a century's worth of evidence to see how well this experiment has worked. And just like with problems in macro, empirical arguments of efficacy depend critically on which counterfactual you choose. I think for most people, they grab a counterfactual that goes something like this: private, for-profit schools will have a strong incentive to cut corners when serving low-income students, resulting in underpaid, underqualified teachers, poor occupational safety standards, and inadequate nutrition. If this is your counterfactual, the low-cost solution is state provision. If your counterfactual is that even low-income parents have a strong interest in the welfare of their kids and will insist on accountability, then the case for government provision weakens. I suppose it ends up pivoting on the vision folks have of their neighbors. Is the median voter closer to Friedman or closer to Sunstein?
Is there an actual market failure? That's not an easy question to answer. Is there a perceived market failure? Yep, and when it comes to the cheap talk inside a voting booth, that's good enough. It'd be nice to have betting markets adjudicate here, but I'm willing to bet that won't happen any time soon.