I suppose the question then would be if the very protocols that were used to defend the United States on 9/11 could create the risk of a nuclear war due to Soviet misinterpretation. Any attacks against strategic assets would escalate things on the American side, but the Soviets might be even more concerned.
From the Soviet perspective, all they would see is the Emergency Broadcast System going off, interceptors swarming from ADC bases, the command and control aircraft going up, and possibly SAC going airborne as well. All United States military bases would go to DEFCON 3, and be ordered to standby for DEFCON 2. If the October 1973 War is any indication of Cold War thinking, SAC might actually go straight to DEFCON 2.
All of that would be coming out of the blue. 9/11 itself is considered a black swan event, but all of those signals going off in a Cold War environment would be an even larger black swan. This would likely be Able Archer 83 scale, if not worse, as it would involve a surprise and sustained increase to the alert level with observable actions in the United States itself. It isn't just a cabinet level exercise.