Wrong. Societies have dealt with far worse situations and managed to survive. Societal cohesion is what prevents anarchy in the long run, not avoiding natural disasters.
The Black Death was pretty horrible and yet European society still exists.
Really though I can’t imagine what you think this pedantic obsession with whether or not everyone gets blown up immediately is adding to the conversation.
We'd be 10 thousand not 10 billion and life would suck but we would survive. And the fallout would clear within a few decades or centuries.
10 000 people is not nearly enough for a modern division of labour - we'd be reverted back to prehistoric technology levels within a generation, with a bonus of fallout and absolute climate dysfunction. Given a sufficiently destructive nuclear exchange literally every single human could die.
The Black Plague is not even a cold compared to what we can engineer.
"Societal cohesion" doesn't do too much to help us survive crop failures and wet bulb temperatures over 35C.
Aside from that, I think you might misunderstand your causal arrows a bit. It is fair more likely that massive industrialization and late state capitalism led to a society that valued the individual over the social unit, rather than that a sudden desire to be individuals sprung up in people's hearts and made a mess of the world.