All right, now for my usual analysis of how democracies fail, applied to the reign of the Thirty. It doesn't fit all that well.
Extreme polarization and strife: The Thirty were imposed by foreign conquest, and to that extent cannot be considered the result of domestic polarization and strife. But even when a democracy is overthrown by foreign invaders, the question remains of how much collaboration and support the invaders have. Some Athenians had gone into exile following the failed coup by the 400. Most of these went over directly to the Spartans. During the city's final days, under siege and facing mass starvation, no one seems to have directly favored the besiegers, but there was considerable strife over whether to make peace and on what terms, to the extent that one man was imprisoned for proposing that Athens accept the Spartan terms, and later, when anything better was clearly hopeless and any peace was better than continued starvation, a few bitter-enders continued to hold out, such that Cleophon was executed and two generals arrested for opposing the terms. Further evidence that there was a non-trivial degree of support for an oligarchy among prospective oligarchs are that, according to Lysias, a majority of the Council before surrender went on to hold seats under the Thirty, and that at least one member of the Council went on to be one of the Thirty and another to be their chief executioner. Add to that the role Xenophon ascribed to the cavalry suggests that enough Athenians favored an oligarchy that we can fairly say the Thirty were aided by polarization. Obviously, most of the oligarchs had in mind something a good deal milder and more respectful of law than what Critias established. Nonetheless, according to my modern footnotes (citing a source I was unable to find), even after all the horrors of the Thirty, Critias' friends build a monument to him, showing Oligarchy putting a torch to Democracy, with the inscription, "This memorial is for the gallant gentlemen who briefly checked the insolence of the Athenian mob."
Abandonment of procedural norms. Athenian procedural norms had been crumbling for some time. They weakened themselves every time a general was put on trial for losing a battle. They took a major hit in the moral panic that followed the vandalism of the herms. They were further weakened when Anytus bribed a jury (unjust though the accusations against him were), And they took a devastating blow when the Assembly sentenced six generals to death in a single vote. Seen in this light, the execution of Cleophon and the arrest of Strombichides and Dionysodorus may be seen as coming late to the party, and desperate measure to prevent the whole city from dying of starvation. In short, procedural norms were undermined, but this does not seem to have been a major factor.
Political violence. This clearly attended the coup by the 400. Domestic political violence does not appear to have been a significant factor in the Thirty coming to power, although they obviously used a lot of it once they were in power. Rather, it was the threat by the Spartan forced the persuaded Athens to cede power to the Thirty.
Paramilitaries. The Thirty appear to have used a paramilitary (young men noted for their audacity, armed with daggers) to intimidate the Council into executing Theramenes. Aristotle says that they were initially supported by 300 retainers carrying whips, which sounds very much like a paramilitary. The Thirty arrested metics in person, accompanied by some sort of attendants who might be considered a paramilitary. And they forced ordinary citizens into taking part in the arrests. But the takeover had nothing to do with private paramilitaries and everything to do with the presence of the Spartan army and navy. And their power appears to have rested on the regular military, reduced to the cavalry and a few select hoplites, backed by the Spartan garrison. (Aristotle places the Spartan garrison after the execution of Theramenes and the beginning of the Phyle revolt, but all other sources disagree).
The danger lies on the right. Assuming one treats oligarchy as right-wing and populism as "left" (as seems reasonably in Classical Antiquity), this fits. It was the prospective oligarchs, not just the Thirty, but their supporters among the cavalry, who wanted an oligarchy. This is not to let the Athenian "left" off by any means. They blundered into the Sicilian expedition, terrorized Athens' leading citizens in the wake of the herm mutilation, executed six generals by a single vote of the Assembly, and refused peace when they had the opportunity for it. But they were not threatening to dispossess the upper classes. Except in the moral panic over the herms, which was fully squelched when Andocides came forward, they did not punch up in a way the oligarchy even pretended to see as threatening. Which leads to the next point.
Fear is a greater danger than ambition. This does not appear to have been the case in Ancient Greece. There is nothing to suggest that the Thirty or their supporters were acting out of fear. Athens was beaten. No radical reforms were being proposed. The democracy had not done anything even remotely threatening to the oligarchy since the herm panic ten years earlier. Ambition, not fear, was the motivating factor.
The importance of a middle class. Does not appear to have been a factor, nor was the issue of radical reforms.
The importance of understanding political parties and a loyal opposition. Political parties in the modern sense did not exist in Classical times, but the Athenians appear to have been beginning, at least, to understand that long-term disagreement was normal, and that opposing politicians could exist within the city, each intending the city's best interests, and not needing to fight. The practice ostracism (banishing the loser of a political dispute) was used for the final time in 415 BC, ten years before the coup, and had not been used for 25 years before that. This was not a case of party strife tearing a community asunder because no one understood that it was normal. In fact, if anyone was unable to tolerate disagreement and party strife, it was the Thirty, hence the deadly split between Critias and Theramenes.
Traits of a right-wing failure of democracy:
Driven by fear, a middle class being squeezed out, and inability to tell radical from moderate reformers: None of these appear to have been at work. The oligarchy simply decided that the democracy had failed altogether and the rightful rulers were seizing power. Critias' motives were even more self serving. He wanted power; others stood in his way; the others must go.
Not dependent on a charismatic leader: It is hard to say whether Critias was a charismatic leader. Certainly, although he was not a sole dictator, he was a strong leader and not first among equals.
Triggered by economic crisis or military defeat: Actually, the Thirty coming to power could not strictly be said to have been "triggered" by anything, since that suggests a strong underlying predisposition in that direction. The Thirty were not "triggered" by military defeat in the same way that the 400 were; they were installed by an invading army.
Conclusion: Once again, whatever its merits in modern times, and even in Ancient Rome, my model just doesn't seem to work in Ancient Greece. Oligarchy was seen as much more legitimate then than it is now. As a result, oligarchs did not just lash out in fear when they saw their prerogatives threatened; they acted openly out of desire for power and the belief that oligarchy was morally superior to democracy. The reign of the Thirty put an end to oligarchic plots in Athens precisely because its casualties included the oligarchs' claim to any sort of moral authority.