MAD alone doesn't stop the race. The arsenal the US and USSR had in the 60s was already enough for MAD to work. Despite that, they spent the next 20-30 years developing even more powerful and ever more insane WMD ideas. Each step forward made the balance more fragile. Each step forward was also another roll of a dice, whether or not it'll be a leap too far, forcing the other side into preemptive first strike.