For schools, it's obvious really. I mean education is the cornerstone of every person's career, if you make it private and completely dependable on how much money your parents make, that's not only completely unfair by just about any ethical standard imagineable, but also simply dumb from an economical standpoint, as a lot of pretty smart & talented people would simply end up not receiving any adequate education simply because of their family's background (not that this isn't a problem already to some extent), and as the average qualification of a large portion of society would drop dramitically.
It would work alright for the top ~20%, and very well for the top 1%, but also only initially until the economy in general stagnates due to a fucking awful batch of human capital. I'm actually astonished that I have to explain this, I've never heard anyone, not even the most anti-state people I know say that we should privatise the fucking school system.
Now for healthcare, this one is a bit less straightforward, and also more debateable really (mostly because there shouldn't be any debate at all for the educational system). Firstly however, single payer healthcare is the only realistic way to insure virtually everyone in a society, which in a state as affluent as the US or any first world country should be the goal anyways, for purely ethical reasons alone. But that's not what you asked for, so I'd rather point to the fact that the cost/benefit calculations of treatments have actually been shown to work significantly less efficient with private healthcare than with private schemes.
Secondly, health care is one of the prime examples of a merit good (ironically along with education), and I guess you're familiar with positive externalities and such, so I won't have to explain this one.
Thirdly, bureaucracy costs are in reality lower in most cases with singular public providers than with private insurances.