A full-out nuclear exchange would be thousands of warheads. It would be all major and medium-sized cities. The radioactive fallout and nuclear winter would kill billions more. There'd be a long tail as Earth's ecosystems collapse, but yeah, there are high-confidence projections that such a war really would kill all living humans.

Around 2,500 nuclear weapons have been detonated in tests[1] and we're still here. The two cities that have been nuked were rebuilt.

An all-out nuclear exchange would kill a lot of people and send humanity back to the early industrial or pre-industrial era, but it's not an extinction risk.

1. https://en.wikipedia.org/wiki/List_of_nuclear_weapons_tests

Humans have survived thousands of volcano eruptions, but I suspect we'd be in worse shape if thousands of volcanoes erupted all at once.

> Around 2,500 nuclear weapons have been detonated in tests[1]

The vast majority of those were underground, and they were not all at the same time.

You're making the false assumption that the goal of the cold war was to kill all living humans. It's a war not a movie plot. The primary goal is to eliminate an enemy country and the most effective way is by destroying their military including all the nuclear warheads they posses instead of shooting some remote french colony island with zero military power that is 5000km away from the mainland.