Russia and US have 1600 deployed warheads each, ready to fly any minute. I don't understand why most people don't consider this as threat. We are living with a gun to our heads.
Don't forget Israel. They'll launch their nukes indiscriminately if they lose control of their region. Their nukes are also distributed around the world on submarines ready to lay waste to major cities the world over. No different than Russia, China, and the USA.
Relative to the Cold War, where biggest threat was the paranoid Soviet leadership overreacting to their expectation of a first strike by the West, there don't seem to be that many scenarios where a war seems likely.
I wouldn't be too sure about that, the Cold War wasn't a one-sided affair where only the Soviets overreacted, that was a common theme for both sides due to the stakes at play.
I also wouldn't say a scenario like that has become less likely, depending on how you look at it, especially in the context of US nuclear force modernization [0], it could actually become much more likely.
Because irrational people exist everywhere and I have no doubt there are plenty of people around, some of them probably in very powerful positions, who are still busy "wargaming" a thermonuclear scenario in which they could end up as "winners".
I don't think the threat level changed greatly, the most likely scenario is a mistake launch and crazy officer. Do you know Russia has a system of automatic retaliation attack?
As for the war threat - both sides had their hotheads during the Cold War, but in the end they were overruled.