First, he opens by citing statistics on how badly our public schools are doing --- "12% of graduating seniors were 'proficient'", and "[g]lobal rankings place our seniors 19th among 21 surveyed countries" --- but neglects to consider a glaringly obvious fact: all the countries that do better rely on public education systems. I can't access Pamela Winnick's Weekly Standard article, which Gelernter cites, but the US Dept. of Education's Trends in International Mathematics and Science Study results for 2003 may be revealing. Students in Singapore, Taiwan, and South Korea aren't going to charter schools.* But you shouldn't need recent statistics to realize this; a basic familiarity with history suffices to recognize that public education is one of the cornerstones of the modern nation-state, and common to all advanced industrialized economies. If some nations are doing better than America, then one should study their public education systems for features that could be adopted in our own, not scrap the entire system.
Second, and equally shamefully, the editorial pulls a huge rhetorical bait and switch. Gelernter opens the editorial complaining about the quality of public school education, but promptly switches to a claim that schools no longer "speak for the broad middle ground of American life" --- which, if you translate from bland, homiletic conserva-speak into the sole example that Gelernter gives, means that public schools teach tolerance towards gays.
Let's leave aside Gelernter's regrettable intolerance, and consider the bait-and-switch. If, as Gelernter claims, "public schools have a right to exist insofar as they express a shared public view of education", then educational quality does not matter. A public school that produces excellent educational outcomes but fails to "express a shared public view of education" still has no right, by Gelernter's argument, to exist. So complaining about the quality of public schools is just misdirection. But Gelernter's editorial would strike the average reader as far less convincing if he just stated this outright, so he must first engage in some rhetorical prestidigitation about educational quality.
So, then, what of Gelernter's argument? Well, basically, I think the language and logic of "rights" that Gelernter uses is pretty unhelpful when considering whether public schools should exist. "Rights" can be a useful framework for thinking about entities that, like human beings, are ends-in-themselves, to be protected from certain harms or guaranteed certain goods simply by virtue of what they are. But things like schools, armies, or corporations are not ends-in-themselves; they're instrumental, existing for some external purpose, and the proper way of justifying their existence is to consider their effectiveness at their mission relative to their costs.
Therefore, public schools should exist insofar as they produce better educational outcomes, on average and in the long run, than if they did not exist. All available evidence, across all known space and time (i.e., between nations, and through history), suggests that some form of universal education greatly benefits everybody, and that public education can provide universal education. Call this the public education theory, which merits that name because it has been thoroughly validated. Some have also suggested a hypothesis --- call this the private school choice hypothesis --- that private schools with vouchers or some other parental-choice mechanism would also lead to adequate universal education, and possibly some improvements over public schools. This hypothesis hasn't been evaluated, but it's probably worth experimenting with.
For what it's worth, in my opinion K-12 education in America is clearly a huge mess, but its problems stem from two fundamental facts that have nothing to do with the fact that it's government-run. First, unlike private schools, public schools attempt to educate everybody, including the students who don't care and the students whose parents don't care --- and in America, those two categories include a lot of students. Second, unlike private schools, public schools largely lack the funds to attract first-class people into the teaching profession. Any privatized school system that had to educate everybody, using the amount of money that K-12 education currently receives, would suffer from similar problems. These are, of course, only hypotheses, which is why it's probably worth experimenting with some form of school choice on a small scale.
p.s. One other, lesser quibble with Gelernter's piece: Gelernter cites the "1910" Encyclopedia Brittanica's claim that "The great mass of the American people are in entire agreement as to the principles which should control public education", and expects us to believe it. First of all, this is the kind of vague and sweeping generalization that anyone with an ounce of critical sense should regard skeptically. Furthermore, as any Wikipedian knows, 11th ed. of Brittanica, like every edition of Brittanica including the present one, was full of errors. Moreover, in the absence of modern polling techniques (Gallup's scientific polling revolution didn't happen until mid-century), how exactly was the author even to know what the "great mass of the American people" were in "entire agreement" about? Lastly, does Gelernter really believe that in 1911, when there were still living Civil War veterans and women were still struggling for suffrage, that there were no deeply divisive social issues? Or does he simply believe that no public school taught about these issues in a fashion that was offensive to somebody?
* Actually, IMO, based on stories I hear from my family, the "success" of students in East Asian countries can be attributed to cultural factors that would be incredibly hard to replicate here. And I put "success" in scare quotes because I actually don't think that the educational system of South Korea, for example, ultimately shapes minds properly.