Why is compromise seen as weakness in so many cultures?

This series made me question the true motivation of many politicians. It seems to me, that any kind of compromise has been seen as a defeat by most leaders over the last 100 years. I truly wonder how that came to be... It was very evident in the depiction of the cuban missile crisis. Wouldn´t it have been awesome if Kennedy and Khrushchev stepped in front of the cameras and said: "Listen people, both of our countries made mistakes in the past. Russia will remove its weapons from Cuba and we will remove our weapons from Turkey. After that, we will work together to make sure that a situation like this can never happen again." Would the American people have reacted in a negative way? I certainly wouldn´t have. A president who shows so much understanding for other cultures and is willing to admit mistakes, would get my vote for sure. There were many other examples in Oliver Stone´s series, where a crisis could have been avoided by simply staying calm and finding a copromise. What were these politicians afraid of? Was it really public opinion or was it always the anger of some right-wing rivals (or maybe even the military as was implied in one of the episodes)?