What causal role does liberalism play in social stability? This is clearly an empiric question but one that has profound normative consequences. It’s also one that can be explored theoretically from the arm chair. And while empiric data might be better, there is no guarantee of this owning to possible confounders and difficulty of measuring political concepts like liberty.
One claim that seems plausible enough is that homogeneous societies are easier to rule than heterogenous ones. The more diverse a society, the more likely there is going to be disagreement on how society should be organised and run. And when there is a disconnect between the rulers and the ruled, this problem becomes even more pressing.
The nature of the diversity is also a key factor here. A society might have intractable disagreement concerning certain preferences or matters of taste. We could perhaps split a community cleanly into parts, solely based on their preference of vanilla vs chocolate parfait, but so long as there is a high degree of consensus pertaining to fundamental political matters, then this kind of diversity is irrelevant.
As long as there is alignment between the rulers and the ruled, then a society will be easy, or at least easier, to manage. Given sufficient homogeneity within society, including between elites and the rest of the population, questions of liberalism vs illiberalism become irrelevant for stability. If everyone holds the same views for example, say that there is only one true god, or about certain sexual practices, then freedom to hold those views becomes a mute point. Whether or not legal protections for certain beliefs and practices exist, those protections are unnecessary because everyone freely upholds certain values. Those who wield power simply do not need to coerce the population when the population willingly complies.
But as diversity increases, whether within society or between the rulers and the ruled, legal protections begin to matter and will, I claim, play a causal role in social stability. As diversity increases, so does the need for legal protections to ensure the compatibility of divergent beliefs and practices. Attempts to mandate practices in diverse societies will therefore lead to political, and possible physical conflict. Diverse liberal societies will be, analytically, more stable than diverse illiberal ones for the simple reason that liberal policies reduce the likelihood of societal friction.
One confounding problem here however, is that illiberal societies are less likely to be as diverse as liberal ones. The very tolerance of belief and practice that characterise liberal societies also promotes and engenders greater degrees of diversity of beliefs and practices. Illiberal societies, with their perchance for coercion and indoctrination, tend to snuff out the flame of any threatening diversity early on. As such, it is plausible that diversity and intolerance within illiberal societies could maintain a high degree of homeostasis for extended periods of time.
Liberal societies by contrast, engender diversity like a kind of social entropy. Homeostasis between social diversity and tolerance (both legal and cultural) is also possible, providing that diversity does not exceed tolerance. But if the pressures of diversity exceed the social capacity for tolerance, then only illiberal constraints to diversity can promote stability, and society becomes less liberal.
So it seems clear then that illiberal but diverse societies will require more coercion to manage than diverse liberal societies and will therefore be more prone to instability. But whether or not illiberal homogenous societies are less stable than liberal ones seems like a question that can only be answered empirically.