It's not vulnerability to western progressivism (which isn't taken seriously on a academic level) but postmodern or poststructuralist critique, which authoritarian states are still privy both as a condition in their general societies in depravity and as exposing epistemic flaws in their narratives.
Is authoritarianism actually susceptible to postmodern/poststructuralist critique?
The philosophcial coherence of postmodernism and poststructuralism is very much open to question.
But even if we grant that they do have something coherent to say, does it actually undermine authoritarianism? Consider for example Foucault’s theory of power-knowledge-Foucault wanted to use it to serve “liberatory” ends, but isn’t it in itself a neutral force which can be wielded to serve whatever end you wish? Foucault himself demonstrated this when he came out in support of Iran’s Islamic Revolution. And are Derrida or Deleuze or Baudrillard or whoever’s theories ultimately any different?
Xi and Putin and Khamenei and friends have real threats to worry about - but I struggle to take seriously the idea that postmodernism/poststructuralism is one of them.
Postmodernism itself by its nature is slippery to define, but it's functional result is endless, ontological deconstruction which is corrosive to any ideology reliant on grand narratives, just as to liberalism, authoritarianism and "historical continuity" is no exception. To properly "defend" yourself against it requires certain ontological structures that are fundamentally at odds with the authoritarian worldview, partly because authoritarianism is quite postmodern. They use it to attack liberalism, but at semantic level they aren't any better protected.
Furthermore on the more real side of thing, the postmodern condition is precisely what many authoritarians, namely China are wary of, yet it's probably true that the postmodern condition has already entered Chinese society with degrading social trust, increasing atomization, excessive materialism, influencers running amok, "bread and circuses" with gacha addiction - everything they critique of liberalism at a social level has come to them regardless.
Your invocation of the “postmodern condition” appears to be conflating the culture of late capitalism with a specific philosophical school which purports to explain it - you can affirm the reality of that condition in the present without agreeing with the proposed philosophical explanation - and does that philosophy actually have any useful response to it, when all it can do is state the obvious fact that it exists, albeit in an obscurantist way? Coming back to Kiryas Joel-maybe that is a more interesting response in proving that an alternative actually is possible-although it is unclear whether the CCP has the capacity to pivot in that general direction.
And that’s the other point - Kiryas Joel is full of grand metanarratives, and postmodern attempts to deconstruct them achieve nothing - nobody is listening. I doubt deconstruction is intellectually coherent - but even if I’m wrong and it is, how is it practically relevant? At present growth rates, Kiryas Joel’s population doubles in less than a decade - will that be sustainable in the long haul? Well, we shall see - but I feel confident in saying that whether it is sustainable or not, has nothing to do with postmodernism or poststructuralism
Pointing to a small town as opposed to superpowers of hundreds of millions is a bad comparison, they are fundamentally different beasts.
Whether postmodernism is coherent by itself dosen't mean much either in its potency to deconstruct, it was already quite effective in destroying the Western "myth", I don't see how the CCP's own narratives are more resilient when they have even weaker assumptions. It's not about listening to them after all, but not listening to the dominant narrative.