I think the threat of nuclear war was much closer at times during the Cold War. US and Soviet forces in some cases were directly shooting at each other in the skies over Korea and Vietnam, for example. The Cuban Missile Crisis was viewed as an existential threat by the US at the time, not unlike how the Russians supposedly feel about Ukraine joining NATO.
People running around scared that Putin will nuke us is pretty much exactly why he made those threats. In his mind, it strengthens his position at the bargaining table. Ukraine agrees to a "peace deal" and the nuclear threats ease, is the concept. That and the oil are the two real bargaining chips Russia has with the West. Kim does the same thing with his nukes/missile launches to gain concessions from the West. Old play from the book.