Thanks for your response.
I agree that education has a private dimension, i.e. that a private one is education, the private one does get a private benefit out of it. But I'm sure you'll agree that education also has a large public dimension. If the private members of a society are not education, they may be more violent, less productive, participate in a less informed way in elections, etc. - the list is long, and I'm sure you know it better than I do. These are all consequences that have a huge impact on society. Now I don't know whether the ratio of public/private in education is 60/40, 90/10, or 50/50, but again, I wonder whether this is relevant at all. What is certain is that the public dimension of education is so high (at least in absolute, and in my view, relatively as well), that it fully makes sense that societies take their decisions in their hands and decide to provide free primary education - and don't leave such a crucial part of society (as crucial as is an army, police, etc.) to the market.
Besides, there is no clear evidence to prove that it would be more efficient.
But this is for the consequentialist argument. There are also plently of principled-based arguments to support a public, free, mandatory education system, not least rights-based arguments. Of course, all arguments have to be wieghed, considered together. But similarly as making a principle based argument in favour of public education should not be considered alone, getting children to learn cannot be considered out of the context and other arguments that are made (plus, I don't see why children would learn more in private than in public schools, but that brings us back to the discussion above...). Yet, you don't mention any of those other arguments and necessary nuances (for the least) they bring to the debate. I know it's a short blog post, but I think it may still be careful about not presenting the economist argument as being the absolute one.