The nuclear football is just for show really. If it comes down to it, a president saying no to nuclear war won't stop the nukes from flying.

I'm reading a book on this atm and I've only read about the early Cold War so far, but for instance, it was widely believed among the military involved with nuclear weapons that Eisenhower had delegated authority to nuke to a bunch of generals under some very lax circumstances that occurred daily like loss of communication (which happened super often due to weather back then), and these generals then did the same with their subordinates. Nobody knew if that authorization was actually real, but they all acted like it was. In essence, you had very decentralized authority to nuke the Soviets/Chinese (US policy lumped them together even in case the Chinese would stay out of a war with the US). I doubt much has changed, perhaps others can comment