Last week, Barack Obama’s administration bought $70,000 worth of airtime on seven Pakistani television networks to broadcast a commercial denouncing the now-infamous YouTube video that portrayed the Prophet Muhammad as a pedophile and a lover of goats.

The video, which set off deadly riots across the Islamic world, is an absurdity — a risible script, bad lighting and pornography-quality acting. But U.S. officials think that the film, apparently produced by a Coptic Christian in California, is at the root of the latest eruption of what has come to be known as “Muslim rage,” and so they thought it necessary to produce a response video.

The administration’s video featured a well-lit President and secretary of state using many judgmental adjectives to denounce the YouTube film and to deny strenuously that the U.S. government had anything to do with its making.

Last week, thousands of Pakistanis, either the sort who fast-forward through commercials or the sort who are unmoved by U.S. propaganda efforts, laid siege to American diplomatic outposts as a way of displaying their religious feeling. At least 15 Pakistanis died in the mayhem.

In retrospect, that $70,000 didn’t represent a wise allocation of American taxpayer resources. It also showed that the administration has misdiagnosed the problems it faces in the Muslim world — and has missed a crucial chance to take more substantive steps to redress them.

In assessing the U.S. response, it’s important to remember that these outbreaks of rage aren’t actually about the YouTube video. Yes, some Muslims — members of the “If-you-don’t-call- Islam-a-religion-of-peace-we’ll-slaughter-you” school of interreligious understanding — react in brittle and aggressive ways to insults directed against their faith, and in so doing humiliate their prophet in ways anti-Islam polemicists could never hope to achieve. But they represent a small minority of Muslims.

Motivation for rioting differed from country to country, but there are common threads. Many of the riots took place in countries with poor economies and venal, incompetent governments with mythomaniacal worldviews. (Recall that the president of Egypt is a Sept. 11 Truther.)

More to the point, much of the rioting could be attributed to the exploitation of religious sentiment by radicals affiliated with Salafism, the extreme, puritanical, anti-Western and anti-Semitic strain of political Islam from which al-Qaida draws much of its ideology. Salafists are competing with secularists and more moderate Islamists for power (only Salafists could make the Muslim Brotherhood appear moderate), and so they look for any opportunity to highlight their anti-American bona fides. This video, like the Danish cartoons mocking Muhammad that set off protests in 2005, was merely an excuse.

So why won’t the administration acknowledge this fact? Because that would mean acknowledging that the killing of Osama bin Laden and the withdrawal of troops from Iraq didn’t bring to an end the unhappy U.S. entanglements in the Middle East. It would mean acknowledging that Obama hasn’t charmed radical Islam into submission, and that American counterterrorism policies, especially drone strikes, sometimes cause as many problems as they solve. It would mean acknowledging that the aftermath of the Arab Spring is messy and ambiguous, and that anti-American resentment in the Muslim world is often a byproduct of deeper dysfunctions of culture, religion and politics. We are sometimes vain enough to believe that hatred of the sort we see on the streets of Cairo, Karachi and Tunis is wholly about us.

Given the obvious truth that this latest spasm of (ostensibly) blasphemy-induced rioting won’t be the last, I think that Obama and Secretary of State Hillary Clinton squandered an opportunity to treat Muslims like thinking adults and to advance a core American interest: the spread of freedom.

Blasphemy, we have come to learn, is taken quite seriously by Muslims. Free speech, however, is taken quite seriously by Americans. It would have been bracing for the President to go on Pakistani television, to sit for interviews with Egyptian and Tunisian journalists, and stand up for a core American principle. Imagine a speech in which Obama described the mechanics of free speech and the undergirding philosophy that protects it. He could have spoken about the great gifts free speech bestows on a society. He could have spoken about how he himself is attacked mercilessly by a free press, yet he still values the principles that allow him to be attacked. He could have described how Christianity is often the target of attack, yet survives and thrives in the United States.

This wouldn’t have been an easy message to deliver. As Hussein Ibish of the American Task Force on Palestine told me, many of these protesters simply can’t fathom the existence of a political system in which the government has no control over the news media, or over what gets posted on the Internet.

Would it work? It wouldn’t change the minds of Salafists, and al-Qaida would continue to seek to kill Americans, whether or not some among us continue making idiotic anti-Muhammad videos. But a bold, uncompromising and guilt-free defense of free speech might have given comfort to the many Muslims, religious and secular alike, who want to lead their lives free of the fear of fundamentalist tyranny, and who would prefer the U.S. not attempt to reason with the mob.

Jeffrey Goldberg, a national correspondent for The Atlantic, is a Bloomberg View columnist.