No nuclear state needs to spend a dime on conventional defence. It's just not needed. The modern defence force isn't for defence.
So you would exploit MAD? When are you willing to use nukes? If you have no military force then your opponent can push you up to that point and you have no other way to retaliate. Even after that you cannot win, just ensure that everyone loses.
And how many nukes would you have? If the enemy can neutralize your nukes you are powerless (Which means that you will need forces to guard them, or submarines that are constantly moving, which is also expensive.) On the other hand, the more you have the bigger are the chances of them falling in to wrong hands. And who will have the authority to launch them? What if that person is not willing to start WW3? What if that person has been neutralized? And again, spreading the lauch codes to several people is just another security risk.
And finally, what if your opponent doesn't have a homeland that you can threaten with nukes? What about rebellion? Show them who's the boss by nuking yourself? Terrorist? Nuking your home is just helping them.
