It was a doctrine that was put into effect. The arms race in terms of nuclear warhead stockpiles persisted because each side had to maintain the ability to destory the otherside's second-strike capability.

I also think that the "Cold War" is a bit of a misnomer. As you pointed out, things got pretty warm. :) I also think you made exactly the point of the doctrine.

they did both escalate their arms and came dangerously close too many time.

I think the point is that they never did, and MAD is quite likely the reason why.