Life in those days was very precarious. One bad drought and everyone would have been dead.
A nuclear war is not likely to kill everyone. Plus a nuclear war is not likely these days anyway.
The chance of a nuclear war today is far far lower than some disease in the middle ages, or a drought in pre-history.
Everyone focuses on nuclear weapons, but reality is they are very very unlikely to ever be used, plus illness is a much more severe danger. And the good thing about today is that we are much better prepared to handle a disease than at any other time in history.
And that's why we are less likely to destroy ourself now than at any other time.