Thermo-nuclear weapons and delivery means were then advancing and the nature of WWIII as a hypothetical event depended on the postulated time. His models tracked this. For the first time I saw why first strike capabilities and civil defense were destabilizing and why military espionage was stabilizing. He plotted a path thru the thicket that left a margin of safety; the path included building many weapons. The logic of MAD preceded the term. From the psychological standpoint I think he made it possible to visualize an outcome without WWIII. Some at Livermore doubted any such possible path; some of those, I think were convinced by Kahn.
Neither Livermore, nor Los Alamos were the center of military strategy but there were high bandwidth channels for ideas between these development centers and military strategy. Kahn’s ideas were honed by his contacts with the various players.
I don’t know what the Soviets were thinking but the mostly public nature of Kahn’s work was presumably known to the Soviets. I do not know of any Soviet counterparts to Kahn. Kahn modeled Soviet strategy just as game theory preaches. The best I can see is that they got the message.
I suspect that before the memes such as produced by Kahn, those who dared to hope to avoid the holocaust were deemed fuzzyheaded, peaceniks or hippies. Some thought they could steer the world by images of the end of civilization; surely that would bring people to their senses. Game theory showed otherwise and steered the world rationally past the pitfalls. It was not without risk however.