This is what the thinking was during the early part of the cold war, the whole duck and cover era. The decades after that and the knowledge gained during the cold war is what lead to our current fears and justification for focusing on disarmament rather than nuclear war prep.
Whether or not humans end up totally extinct is irrelevant. Enough post nuclear fiction has been produced that people are well aware, yes, we may not all die, but life sure is going to suck a whole lot for literally every living thing on Earth.
That's the point of preventing nuclear war, that's the point of the world's current attitude towards it.