For MAD to be effective you need to convince people you’re both able and willing to vindictively kill off large swaths of the worlds population. As such you do want to come off as just slightly unhinged which having first strike capability projects.
First strike was needed in order to slow things down and do some damage, in theory to a) give the US time to rush conventional forces to Europe, b) bloody the nose of WP enough to make them think twice and c) destroy as much as possible of the WP rear areas and reinforcements. If the WP had gone through the Fulda Gap or the South option near Switzerland -tactical nuclear mines, Special Operations nuclear devices, tactical nuclear artillery, tactical aircraft attacks etc would have been used pretty quickly on them.
In reality it probably would have escalated to far worse than that pretty quickly I'm guessing.
Basically, first-strike was a posture in public. The real question was "who commanded it"
Which is kind of an oral history of the development of the current command and control systems (includes some great notes on things like "how unlikely the air force thought a mid-air collision of nuclear armed bombers was" - the spoiler is surprisingly more likely then it was thought after the second one happened).
https://www.amazon.com/Doomsday-Machine-Confessions-Nuclear-...
Although it's not nuclear war, a better example in the same vein is Special Bulletin.
and here in this PBS documentary: https://www.youtube.com/watch?v=jlPEBROvR9w
Edit 1: I think that's the one (in German): https://www.youtube.com/watch?v=fJ3vkqDaPdg
Edit 2: It is set before the fall of the USSR and the German reunification, classic cold war stuff. Just watching it. Forgot the details, but damn the fall of the iron curtan could really have ended differently. Basically, Gorbatshev was forced to step down and military hawks took over.
Both the French and the English decided to maintain their own nuclear deterrent for this reason.