I understand the idea of MAD, but would the USSR really be willing to engage in all out war with NATO without nuclear weapons?
We really can't know the answer to complex hypotheticals like that. Smaller "What ifs" can sometimes be productive. But "what if the strategic situation of the world was totally different?" isn't something that historians can readily answer.
What they can do is point out that, 1) the Cold War was plenty "hot" if you lived in one of the places consumed by proxy war or state interference; 2) "mutually assured destruction" as a condition did not really apply until the 1960s, arguably the 1970s, because the Soviet Union's nuclear capabilities were far undermatched against the USA's, though the question of whether deterrence broadly (which is not the same thing as MAD) mattered is still up for grabs; and 3) there is evidence that there is more than nuclear deterrence that kept the Soviets from initially pressing westward in the 1940s, and there is clearly more than deterrence with the US lack of interest in engaging the Soviets militarily (or with nuclear arms) prior to the Soviets having a truly compelling nuclear force (e.g. before the mid-1950s if you care primarily about Europe, the 1960s or 1970s if you care more globally). The "nuclear taboo" — lack of use of nuclear weapons — has applied even in non-deterrent, definitely non-MAD, situations.