Yes, the educated benefit from being educated, but everybody benefits from having educated people around. The former is why private schools are seductive to many, but the latter is why we should embrace education as a public good - external to the market - and support/fix our existing socialized system.
So you're right, the problem is the incompetence of public schools. But privatization ain't the solution.
Libertarians, who are often persuasively consistent (and I really do appreciate your consistency), have given monopolies, governments, and other non-market institutions a bad reputation. Even the term for something that doesn't jibe with a market - "an externality" - belittles the importance of things like pollution, basic science, education, overfishing, national defense, a judicial system, national highways, and on and on and on.