This is a very interesting post on a hugely important area. There'd be a lot to comment. Let me just point out to two things.
First, your whole piece is based on the assertion, which you get rid of in one paragraph, that education is "largely" a private good. I don't know whether there is a lack of evidence to show that it is a public good, and I'm sure many empirical researchers - from both the economics and from other fields would dispute that. But then you give the counter-argument to your argument: most governments (and not only in the developing world - in fact, I would say that this is mostly the case in the developed world) assume education is a public good. It may be that it is so obvious that no one even bothers looking for evidence. The public being made of a sum of private entities, if the private entities improve, it is clear that the public will hugely benefit. In fact, the public and the private are so inter-linked, that I'm even wondering what the point is in making such a distinction, and trying to know whether education is 40% a public good or 60% a private good, or conversely, or anything else. Fact is that education benefits so hugely to a society, that societies have, to a huge extent, decided that it was a public matter.
Secondly, the kind of discussion you're having, is mostly based on consequentialist arguments. This is what the title of your post involves (economics matters...), but you regrettably don't mention any more this limit in the post itself. Maximising the value per what-children-learn is a valid approach, but it's not the only one. Thinking about what we consider as education, what is education, what place do we want education to take in our societies, etc., are as valid entry points, which we need to equally consider.
And for those crucial choices - privatising education? making it a private good? -, hearing the voice of the people is essential. You also forget to mention that in most parts of the world, collective choices of people, wherever they were given a choice, has been to keep education a public good. Should your arguments about it being a wrong choice, in terms of economics, even be right, it is not all what matters.
For more information on other approaches, worth looking at other empirical arguments from a more social justice perspective, but more holistically, worth lookin at http://www.periglobal.org/, http://globalinitiative-escr.org/advocacy/privatization-in-education-research-initiative/, and http://www.right-to-education.org/issue-page/privatisation-education.