Define "hot war". There have been plenty of fighting in border regions of India and Pakistan in recent years, and both have nukes. India since 1974 and Pakistan since 1998.

There have been plenty of wars being fought. And even proxy wars between nuclear powers!

But those don’t affect you.

And I’m sure you are 100% convinced that if we could snap our fingers and give all sovereign states nukes then all fighting would stop. So what’s the point.

And if you’re wrong? Then we only risk the permanent destruction of advanced human organization. No biggie.

I would contend that it's really interconnectedness (aka globalization) and the general march of technology (especially long-distance weapons) that have contributed to the (very) relative peace of the post-WWII age, but I will readily admit that your point of view is very reasonable and there is a genuine discussion to be had here.

Chinese troops and Soviet planes were there in Korea.