* I would think that we'd be too afraid to try something like that during the cold war in fear of starting a legitimate war.*
Yes, but the system could be located in advance and disabled as part of a first strike. Even the vague details in the article would give an attacker a lot of ideas on how the system could be disabled -- apart from bombing the bunkers (which is difficult) you could spoof the sensors, or just convince the computer that it's still in contact with Moscow.
And even if the Americans can't find a way to disable the system, the fact that the Russians don't know that the Americans can't disable the system means that the Russians can't trust it any longer and so are back to being more likely to make a first strike.
The optimal solution is probably just to hint to the Americans that you might have such a system. Then again, you also have to assume that the Americans are clever enough to have figured out that you might have such a system anyway, so you probably don't even need to hint about it.
Gosh, Cold War game theory is fun. I wish I could travel back to the 60s and get a job with RAND.