(Neither nuclear war nor climate change are considered as this-century extinction risks by the singularity-minded side of the existential risk community, just because there would necessarily be pockets of unaffected people.)
I'm always a bit surprised at how people seem to wish away the destructiveness of nuclear weapons or the likelihood of their use. Another commenter alluded to a "failure of imagination" which seems very applicable here. You would think just watching videos of above-ground tests would cure that but apparently not. [0]
Oh sorry! I meant that the people to whom you referred above focus on the "one true catastrophe" (whatever that is) without truly thinking what it would mean to have a few dozen nuclear weapons used. It's not just the immediate effect of the explosions. It's also the destruction of power distribution, industrial capacity, food and medicine production, and other things that millions of people depend on.
No worries, but I think we're still dancing around the same point. I think those people would say that they agree that nuclear war is very bad, and that it has many nth-order consequences which are all clearly catastrophic, but that (for example) while disrupted supply of medicine causes a great deal of human suffering and unnecessary deaths, it remains difficult to imagine how it might result in literal human extinction.
The threat of nuclear weapons is massively overblown (pun intended) thanks to a bunch of Cold War fearmongering.
The Earth land surface area is 149 million sq km and there are only about 12,500 nuclear weapons in the world. Even if they were 10 megatons each* and were all detonated, with a severe damage blast radius of 20km (~1250 sq km), it'd cover about a tenth of the land available.
Since the vast majority are designed to maximize the explosive yield, it wouldn't cause the kind of fallout clouds that people imagine, nor would it cause nuclear winter. It'd be brutal to live through (i.e. life expectancy of animals in Chernobyl is 30% lower) but nuclear weapons simple can't cause a human extinction. Not by a longshot.
* As far as I know, no one has any operational nuclear weapons over 2 megatons and the vast majority are in the 10s and 100s of kiloton range, so my back of the napkin math is guaranteed to be 10-100x too high.
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