Well then, let me enlighten you about the United States education system. Sit back and enjoy a bit of history.
Back in the early 20th century, it was discovered that faculty education theory does not hold. This is to say: the brain is not a muscle. Faculty education purports to strengthen the brain by flexing, such that repeated memory drilling makes memory work better, and the learning of Latin and Greek make the mind's ability to process language sharper and firmer. Such things are a complete falsehood.
As reaction, John Dewey lead a progressive education renaissance, replacing this with a sort of student-centric education by experience. Rather than studying from biology textbooks, memorizing times tables, and reciting historical events, students would grow plants from seed, perform mathematics from word problems, and read historical stories and enact plays and games. This would replace "I know" with "I have experienced", creating a life of experiences as education.
There are obvious faults in this. From an unstructured and unknowledgeable view, one could easily comment on Latin and Greek providing the basis of English and other European languages, thus providing strength in English. This argument is fluffy and makes no notable contribution; it is, however, correct for other reasons, which I shall expand upon.
To start with, the ideals of John Dewey imply that "experience" is all-important and, directly and explicitly, damn the concept of "memorization". According to this new progressive education method, memorization is wrong, is toxic, and is damaging. Memorization is painful, difficult, detracting, and dull; it distracts the student from the pleasure and enjoyment of learning by his experiences, and provides no real benefit. This view is patently false.
All learning, all knowledge, predicates on memory. A person cannot learn what he cannot remember. Experience forms a type of memory, and forms it more strongly than rote drilling. I will agree that the rote memorization of facts is dull and toxic; but this is not the only form of memorization, and it is not reason to eschew memorization entire as a damaging or even an unimportant aspect of education.
What faculty learning misinterpreted--and what its detractors completely missed--was the improvement of the mind by *technique*. Memory, for example, improves by mnemonics such as acrostics, the method of Loci, rhymes, and visualization. The learning of Latin and Greek provides a familiar base on which to derive English and other European languages, easing the learning and use of such, at least in terms of vocabulary. These are not matters of the brain getting stronger, but rather of it having knowledge of ways to approach these things more effectively--in the same way that a crowbar does not make you stronger, but it does allow you to rip the side of a house clean off.
The many skills we need for a strong education system include mnemonics, study systems, mental mathematics, problem analysis systems, decision making systems, and the like. I will expand on each in brief here; I will trade a strong and complete argument for brevity and structure.
Mnemonics form the foundation: what is not remembered cannot be known, and what is not known cannot be learned. Mnemonics allow for quick, efficient memory of things, at least temporarily, so that they may be recalled in the course of learning or applying without carrying a reference. Such recall allows learning and doing to apply freshly-introduced concepts which may otherwise be promptly forgotten, which allows for understanding of further explanation, and thus allows for more solid learning. These techniques thus increase learning and thinking efficiency.
Study systems closely relate to mnemonics. SQ3R and SQW3R propose a method for examining new material (largely, written material) and preparing to meet it in learning. The general mode of these and other study systems is to Survey the headings, pictures, introductory sentences and paragraphs, summaries, and other topical aids; Question the material, often by turning these glanced-over elements into questions; Write the questions; Read the material, being mindful of what you know and what you have questioned and gleaned from a quick survey; Recite from memory what you have learned; and Review the material and your notes. In modern schools, we are told to "take notes" and to "study", but not told how to do these things in any useful capacity.
Mental mathematics include all techniques for rapid computation, as we do not have such a term as "mathmonics" or such to describe them. The operation of a Japanese Soroban abacus, and the conduction of its processes in mind rather than in machine; Friendly Number systems; lattice multiplication and Napir's bones; the methods of derivative and integral calculus by pull-down, chain rule, or U-substitution. Structured teaching of such things founds our arithmetic education more strongly than modern education methods, which in turn founds algebra, founding geometry, founding such things as adjusting your baking recipes for two round 9 inch pans rather than an 18x24 rectangular pan.
Problem analysis systems structure the analysis of problems. My favored is the Kepner-Tregoe problem analysis system, which specifies a number of questions (eleven in total) for any performance deviation problem. When a thing is not working, the Problem Specification strategy mandates first identifying What is not working, What it's doing wrong, Where it is, Where on it (what part) is the problem, When the problem first occurred, and so on. It specifies answering for what IS having the problem; what IS NOT but COULD BE experiencing the problem (i.e. two 2004 Honda Accords, one doesn't have oil pressure); what is DIFFERENT between them; and what CHANGED when the problem started (or occurs each time the problem occurs). There are several such systems which provide structured ways of attending such problems, providing for rapid and efficient troubleshooting.
Decision Analysis is a similar concept. Decision making frameworks range from Pugh and Weighted Pugh matrix methods up to the related Kepner-Tregoe Decision Analysis framework (effectively a Weighted Pugh matrix with baseline selected per comparison dimension), and move horizontally to unrelated decision frameworks such as analytical hierarchy. Such frameworks provide for structuring and analyzing decisions between similar things, and reduce debates over alternatives to debates over what is important and how well any given alternative satisfies that need relative to others.
I'm sure you can think of other skills we need in education as a foundation. Mnemonics and Study Systems form the basis of studying anything, while the mathematics systems I've described strengthen our ability to learn and use math. Problem and decision analysis systems are things we would learn further down, as they are useful *functional* concepts but not foundational *learning* concepts. You could easily argue for critical thinking skills--the study of logical fallacies, for example--and for risk analysis skills as part of education of primary importance. All of these things are left out of our standard curriculum; some of them would improve our ability to learn the curriculum, and others would simply improve our decisions in everyday life.
It is these things which make an advanced, enlightened society. These were things the ancient Greeks and Romans founded society on--a society without a printing press, without high levels of technology, without the benefits of modern life, of modern communication, transportation infrastructures, and space travel; yet they built great buildings, running water, fire departments, speaking halls, farm lands, and even empires. Imagine those ancients with the technology we have at their disposal--a people who saw a book once in their life, if they weren't simply confined to hearing another man tell at length its contents from one reading, and incorporated that knowledge fully into their own. We are as naive children; they, however, did not have the scientific process, or the firm grasp of the universe which we have derived with it. Who has made best of the resources they had at the time?
You say a school only needs the free market, books, and a handwave; better education systems have been had for less, and worse have been produced with more. In the proposed world, we would have very little, and would be charged to make a lot of it. What greater thing can we bring to that world than the knowledge of how to make all persons geniuses on the level of Einstein, of Hawking, of Socrates and Plato, of Simonides, of Ben Pridmore, of the bards who memorized the epics of Homer? The disciplinary processes of the mind, the mnemonics, the mathematics, the analysis, the scientific process itself, are what allow a man to turn rocks and mud into concrete and steel, into high-rise buildings, into space ships and nuclear power; it is those processes which we would best bring.