Nuclear war will not be enough to cause human extinction. The targets are likely to be focused on nuclear powers which leaves many areas of the world untouched: e.g. South America and Africa. Life will definitely be quite unpleasant for the remaining humans but it will not cause the world population to drop to 0.
I am much more concerned about biological weapons which do have the potential to cause absolute human extinction.
For example, some total whack job of an authoritarian leader is in charge of a sufficient nuclear arsenal and decides to intimidate an adversary by destroying a couple minor cities, and the situation escalates badly. (stupidity)
Or we finally pollute our air and/or water with a persistent substance that either greatly reduces human life span or reproduction rate. (hubris)
I think either of the above is more likely to occur, and I am not commenting on current world events in any way. I think when something bad finally happens, it is going to come completely out of left field. Dr Strangelove style.
And the last of us will be saying "Hmmm, I didn't see that coming".