Well, he's using some polemical trickery, bait and switch and a good bit of nonsense to mix fact with his political views, so point by point:
- Catholicism leads to monarchies, Protestantism to republics, Islam to tyrannies: I just don't really know where to start with this. Italy would be very surprised to learn that it's not a republic. The United Kingdom would quite surprised to learn that it's not a monarchy. I think you would struggle to find evidence the Ottoman empire was somehow particularly tyrannical in an early modern context in comparison with, say, the Russian Empire or any other absolute monarchies. It's entirely possible that this is what Montaigne thought, and in turn what some of the founders concluded from Montaigne, I don't know enough about their political thought to comment on that, but as a socio-religio-political observation if this is what Montaigne thought I don't think he would receive any support in his views today based on the evidence at hand.
- "So from the beginning we knew that Islam was not a religion of peace": Ones mode of selecting government has no bearing on whether or not your religion is one of peace or not. If the implication is that Christianity somehow is a religion of peace despite its selection of government than you would have to account for Christian nations conquering, at one time or another, virtually the entirety of the globe and leading the way in intra-Christian slaughter, between, say, the 30 Years War and VE Day 1945 in a way that was totally unprecedented in the Islamic world.
- "Because it reigned by the sword": All states reign by the sword. That is more or less the most basic definition of a state, as Weber had it a state is defined by the monopoly of the legitimate use of force. Now that being said I actually wouldn't argue that "Islam is a religion of peace" because I think those kinds of phrases are non-sense but insofar as Islam is intricately connected in its doctrines with a state building process then there is going to be a connection, as in all states, between the state's monopoly of violence and the religion's involvement in the state. That's just as true of Christian states, even relatively pacific ones like the Vatican.
- The above being said, I think he's actually alluding to well known phrase that Islam was "spread by the sword" rather than that it "reigned by the sword." The idea that Islam was "spread by the word" in the sense that non-Muslims were everywhere and always converted by force does not have credence among historians of Islam. It's simply untrue, albeit based in part on a misunderstanding about the historical context of some of Islam's own writings that were indicative of a more bellicose Arab conquest than the contemporary evidence suggests.
- "Wherever Islam was dominant you had conflict, you had fights, all you have to do is look at the Islamic nations around the Mediterranean now, see if they're nations of peace" : Did you feel the force of historical whiplash as you ripped through 1400 years of Islamic history to draw conclusions about an entire religion based on contemporary politics? Because I did. Pax Ottomana was a thing. Mass civil conflict in the Middle East for most of its history under Islam was not. Even when you have successions of empires in the medieval period, it just didn't involve a whole lot of mass slaughter. Again, given the scale of conflict in Europe over the same period what exactly are we supposed to be comparing this to?
- "We had to deal with Muslim terrorists attacking Americans" : That's astoundingly bad history. There are a lot of ways you can describe the Barbary Corsairs. "Muslim terrorists" is not one of them. Terrorism is a loosely defined phrase that's often misused but it typically involves some sort of asymmetric attack designed to instill fear in a more powerful adversary as a means of provoking that enemy into making political change that will either even the battlefield in some way or get the enemy to bend themselves to the terrorists' will. "Islamic terrorism" usually implies that this is being done with a religious motivation. The Barbary Corsairs were basically privateers. They wanted money, not a political change in the United States. They were acting on behalf of independent or pseudo-independent states using conventional naval warfare tactics to capture shipping and hold prisoners for ransom (the treatment of those prisoners varied pretty wildly but many of them were in fact well-treated.) Even calling them "Muslims" as a descriptor requires a bit of explanation because many of the Barbary captains were adventuring European converts with, perhaps we should say questionable piety and devotion to the faith.
- "Their quran says they have to attack us": Look, I don't know what this particular letter to the state department contains, I'm not terribly familiar with US revolutionary era foreign policy, and it's entirely possible that whatever diplomat or Bey they were negotiating with did say that, but I can tell you that's emphatically not why the corsairs were attacking. How do I know that? Because the Corsairs were not attacking states that paid them tribute. The reason why this suddenly became an issue for the US is that prior to the revolution, colonial shipping was protected by the British tribute to the Corsairs. And it's why after the revolution the Corsairs only really stirred up trouble with the US.
- "15% of the budget was being spent trying to mollify Islamic radicalists" He's right that huge percentage of the (very small) US federal budget was being paid in this tribute. But "Islamic radicalists" for some of the reasons I mentioned above, is just nonsense here.
- "Just squashing Islamic terrorists": lol, nope, again, not terrorists. The conflict also didn't end with them being squashed. It ended with negotiated treaties for the corsairs to respect US trade. What's more, I wonder what the creator of the video would make of another erstwhile Islamic corsair nation, Morocco, which was the first country to recognize the independent United States.
Verdict: Bad history
Sources: I know he's got political baggage of his own, but I've been reading Max Boot's Savage Wars of Peace which seems to give a much more practical and even handed account of the Barbary Wars.
edit: qualification to the monarchies, republics, tyrannies thing.