Free markets don't just develop innovative forms. They also cater to all tastes from which a profit can be made. With government spreading around a lot of tuition money, that means that means that almost any culture can be served. There will be schools known for their music, drama & art, or their science & math, or their multiculturalism or their eclectic and nondogmatic treatment of history, or their competitive sports teams...
Diversification is the essence of establishing market niches. Businesses set themselves apart so that there's something they provide that others don't. That something, if people will pay for it, is an opportunity to make a profit. Hence, free markets foster variety.
Look at food. It's at least as important as education, since without it we would all die. Similar to education, the government feels an obligation to assure "universal nutrition". Thankfully, it doesn't feel the same compulsion to own and operate the whole food system the way it does with education. If it did, we would probably be eating government surplus milk and cheese three times a day in cafeterias that look like prisons, and "home cooking" would be as controversial as home schooling.
Instead of that nightmare, we have a free market (several actually) starting with farms and proceeding to grocery stores and restaurants. Farms make a bewildering variety of plant and animal products from cheap, hormone driven and pesticide laced stuff to free-range and organic alternatives. Food in the stores runs the gamut from junk snacks and heavily processed foods to organics and staples. For a price, we can buy exotic and out of season fruits and vegetables from other parts of the world. On a budget, we can buy oranges and apples. Like grocery stores, restaurants offer a bewildering variety of selections from a bewildering variety of cuisines.
Think of what you want in your local school that isn't there, or is always on the chopping block. Think of the programs that are there but that you think are a waste of money. Wish you had a fully equipped football team? Have a fully equipped football team and wish the money had been spent on a theater season instead?
Want advanced placement classes for college prep? Or, do you have them and think the money would have been better spent on trades? In a free market, enterprising providers would vie for your education dollars by offering these choices and much, much more than I can imagine.
Besides electives and extracurricular activities, there's also the atmosphere. In what kind of philosophical or religious environment would you immerse your kids? From what religions or philosophies do you want to isolate them? What do you consider virtues to be promoted? What do you see as character flaws to be inhibited?
Is there a particular culture or heritage you would want to fill the reading and history? What holidays do you want to celebrate, and in what fashion? What language(s) do you want your kids to learn, and from what age?
It's a tempting idea, sort of like testing people to see who's competent to vote. Unfortunately, the temptation to abuse the power is so great that I can't think of a way to protect it.
With the government involved, the exam would become a political football filtering for "political correctness" rather than academic excellence. Diversity would suffer, not to mention that it would generate all of the "teaching to the test" problems decried elsewhere.
If standardized testing can raise public confidence in grades or diplomas, then people will establish some kind of national testing service(s) privately, which continued success will depend on credibility, not political favoritism.
Religious fanatics must be willing to let atheists spend tax dollars on sacrilege, and atheists must be willing to let theists spend tax dollars on religion. I can already hear both sides shout, "Never! I don't want my dollars spent on..." I've got news for you all: People of all stripes pay taxes and send kids to school, so it all comes out in the wash.
Just pretend that those folks you don't respect are spending dollars taxed from their own ilk. As long as essential universal education is achieved, then the government and society are getting what they paid for, so it just doesn't matter what is added (as long as they don't endanger the rest of us). The right-wing Christians should be allowed to go to their schools with their limited reading lists, and everyone else should suffer with all of the "corrosive" books that have been banned.
Likewise, the presentation of history need no longer be redacted to nothingness by competing revisionists compromising on the scant intersection of their various interpretations. Let each history be taught in its full glory. In a free market, with the narrowminded off doing their own things, there might even be a market for schools that teach multiple viewpoints. What a concept!
With just one system in each community, there's just one flavor, and it tends to be a bland compromise, offending many and satisfying few. In a government system, the only way to get what you want is to build a majority coalition to run roughshod over everyone else who will be stuck with it. Judging by the rancor now raging over so many monopolized school systems (the Supreme Court gets about 500 cases per year just on religion in public schools), one family's tonic is another's poison and vice versa.
However, if we had a free market, then any group (market segment) large enough to make trouble today would be able to go off on their own and leave the rest of us in peace. Nobody would have a need to impose on others. Some tyrants might still try to do so, but we could all get together in shouting them down, telling them that since they're free to fill their own kids' heads with whatever nonsense they like, then they must afford us the same courtesy.