Honestly I think a big part of the difference between Cold War propaganda and modern propaganda on the subject is simply time. A large chunk of the people who were so into the idea of conflict with Russia also remembered the day that the Newspaper showed them images of a single bomb wiping a major city off the face of the Earth. People now a days don't have that type of reference. I have a feeling if a city like Kyiv (God forbid) got obliterated by a nuke people would not be so dismissive of the idea of nukes being something to scared of

Edit: I'm just using the city of Kyiv as an example because the idea of it being nuked has regularly been in the news and dismissed as an impossibility by regular people