I disagree, and fail to understand why you think that. Why can a free market education system not provide for the masses? The free market does so with food and clothing. Why can't a fluid free market meet educational demands?
Public education does not offer a "BMW". They offer a Gremlin. The free market will offer the Gremlin as well, but it will also offer everything in between. We'll all be better off, unless "being better off" means people are equally dumb.
I've got the day off and been spending much of my free time as of late reading on this subject, so get ready for a short book.
My point is this: The benefits of equity in attainment are obvious, and need to be achieved. Free markets by their very nature ensure efficiency, not equity.
First, a look at charter schools: Many have hailed its benefits, but the bottom line is that they have by and large shown little to no improvement in educational outcomes, even a sizable percentage (around one fifth IIRC) have actually done worse than their counterpart public schools within the same districts, while the same amount have done better. The middle three fifths tend to do about the same. They may operate more cheaply, but to this point there is no sign that this form of competition actually makes for smarter students. The same has been found of voucher programs.
The original intention of charter schools was to be a cooperative program with public schools to test out new ideas, measure success and implement them in public schools. They have instead turned into competitors.
If I understand you right, public education systems are essentially a monopoly on education and therefore have no incentives for improvement. That is being held up as proof-positive of the failure of public education, in spite of the fact that about three dozen other countries in Asia and Europe have utilized public education, with many cases in which that is the only option, and shown clear, marked improvement. Even broader shifts towards private education, such as those in Chile, have shown no marked improvement in achievement. That alone contradicts the original claim.
The failure of our public school systems can be traced primarily to its origins: A hundred years ago, something like five or six percent of people in the US completed high school. The goal at that time became simply to get most citizens through secondary school, and by an large it worked. Something like 85% of people in the US now graduate high school.
The paradigm has shifted, though, to producing students in large numbers that are ready for college, and policies so far have been ineffective, but there is 100% clear proof that it can be done, because it has been done.
It takes a very basic look at the way competitive a market needs to be organized to function properly, and why the very nature of education throws a huge wrench in the premise. Among the basic needs are a large number of buyers and sellers, a relatively homogeneous product, free entry and exit of firms, lack of transport costs, and independent decision making to ensure profit maximization. Two other major aspects are perfect conditions in information and mobility.
The first problem is treating students as buyers of education and schools as sellers of that product. Students are not buyers here, they are consumers, and those are two distinctly different things. The taxpayers are the buyers of education, and what they are buying are the greater social benefits of having schools in their state and community. If public funding were to be eliminated and students were truly turned into buyers, then the quality of the education one could afford would be tied to their means, resulting in an inherently unequal outcome. As for the fact that students are not buyers, that by itself creates a market distortion. Similarly, schools are not sellers of education, but rather producers of it.
Next, back to the BMWs and Gremlins. Car companies do not exist to ensure universal access to high quality cars. Clothing companies do not exist to ensure universal access to nice clothes. Food companies do not exist to ensure universal access to food. They exist to make profits producing those products to buyers when and where there is demand. When it comes to consumer goods, that is the desired outcome and as such that market form works best. Moreover, they can enter a market as soon as they can make a profit, and they can exit a market as soon as they can't. Imposing this form on schools would not lead to socially desirable outcomes. Not everyone can afford to buy a new car, so they take the bus. Not everyone can afford to feed themselves or their families properly, so there are food stamps (which also explains the presence of food deserts, where access to proper grocery stores is limited or non-existent, and there is a proliferation of corner stores with junk foods). In a free market with perfect conditions for entry and exit of schools, not everybody could afford basic education. Absent a voucher system, schools would not operate in areas where there is low profitability, and if they did, it would be a lesser quality product, while higher quality teachers and staffs would open up shop in areas. There's a reason that BMW dealerships are adjacent to nice neighborhoods, and poor neighborhoods might have a Kia dealer if anything. That's the natural distribution of goods in markets, and is fundamentally incompatible with universal educational achievement.
Next is independent decision making for maximization of profit and utility. This concept has already been breached by the presence of compulsory education. Students, aka consumers of education are already forced to stay in the market longer than some may have chosen to if they were free to do so. Consumption has already been forced into inelasticity. Here again, the very nature of a free market education system proves incompatible with the notion of universal college readiness, or a decent education at the least. If compulsory education stays in place, then that means there is a continuous presence of students whom it would be unprofitable to educate, i.e. special needs, disabled, etc. It's not unlike people with a strong family history of early cardiac failure being unable to purchase health insurance because it becomes practically impossible to insure them and maintain profitability. This ties into the notion of schools as selling a service; in many districts, they are forced to compete with charter schools for increasing the overall student body in order to increase funding. The basis for selling a consumer of education on attending that particular score must be quantitative in nature, which requires the presence of standardized testing to show achievement. I think we're all in relative agreement that this will introduce things like "teaching to the test" which ties into Campbell's law: "The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor." Fundamentally, performance-based funding, or the profit motive in a free market system will wind up occurring in the form of incentive to cheat rather than to truly reform.
As for perfect information and mobility, they get undermined as well. In a perfect free market system, every buyer is aware of the cost and quality of every good the buyer is selling specifically because of the price mechanism, while every seller is aware of the size of a market and demand for what they are selling through that same price mechanism. Simply requiring everyone to go to school completely undermines the pricing mechanism. As for mobility, I hinted at it earlier, it requires all resources and firms to be able to enter and exit markets. This extends to students, specifically, who are relatively immobile.
I'll take a break from typing for a bit, but I read a few reports and a book by Marc Tucker, as well as some serious digging on what reforms have proven to be effective elsewhere in the world.
But, for now, hopefully I've made some sort of impression as to why I believe that a free market education system is fundamentally incompatible with producing high achievement across the board and would ultimately leave us in the dust in the global community.