A few hundred or thousand nuclear bombs dont kill all humans and dont destroy this planet, there will be billions of people who will survive a new world war. Nearly all people in africa, south america asia and so on will survive imo.
Subject's scholars & practitioners seems to disagree with your statement. Studies shows that even a limited nuclear regional war - e.g. India vs Pakistan conflict, a scenario dire enough to have about 20 million people dead due to bomb blasts and subsequent fires and radiation - could have consequences on a global scale due to a global cooling of about 1.25 °C (average) by killing crops & possibly starving about 1 billion people worldwide. A regional nuclear war could cause widespread loss of life even in countries far away from the conflict.