It's pretty nuclear deterrence theory; the major powers both want (1) to actually be survivable and thus free to use any means at their disposal knowing that the other side cannot impose unacceptable costs in return, (2) failing that, appear to the opponent to believe that they are in such a condition, so that the opponent must act as if they will feel free to escalate unbound by fear of retribution even if they rationally should not, and, in any case, (3) prevent the opponent from perceiving (whether or not it is correct) themselves safe from retribution, so the opponent cannot escalate unchecked.
MAD is just the state where both powers acheive #3; it is consistent with either or both powers also acheiving limited success at #2 (which means that policy which doesn't involve nuclear provocation or similar existential threat must be constrained by the risk of nuclear escalation by the other side—if neither side has any success at #2, that concern does not exist.)
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It's pretty nuclear deterrence theory; the major powers both want (1) to actually be survivable and thus free to use any means at their disposal knowing that the other side cannot impose unacceptable costs in return, (2) failing that, appear to the opponent to believe that they are in such a condition, so that the opponent must act as if they will feel free to escalate unbound by fear of retribution even if they rationally should not, and, in any case, (3) prevent the opponent from perceiving (whether or not it is correct) themselves safe from retribution, so the opponent cannot escalate unchecked.
MAD is just the state where both powers acheive #3; it is consistent with either or both powers also acheiving limited success at #2 (which means that policy which doesn't involve nuclear provocation or similar existential threat must be constrained by the risk of nuclear escalation by the other side—if neither side has any success at #2, that concern does not exist.)
It is obvious why the US would want to appear invulnerable. You claimed it tried to. I asked for proof. You didn't provide it.