The 0000000 code was not the football code. It was the Silo Commander's half of the PAL. Arming a nuclear warhead via the PAL system requires two codes - one of which comes from the President, the other of which is a confirmation code from other authorized officials. At DEFCON 2 (War potentially imminent), the commander of individual missile silos was an authorized individual if he had reasonable reason to believe that there was an in-progress attack on the United States. This was no secret - it was a hedge against a decapitation strike that took out all the people who could confirm a strike, and thus publicizing it was a crucial part of maintaining the balance of terror. Allegedly, the Air Force decided that individual silo codes were unnecessary because there was no reasonable way for a hostile actor to take possession of them, and set them all to 00000000 to ensure that there was no possibility that "SHIT, I FORGOT THE CODE" would delay a launch until the destruction of the silo. This has been disputed.
The other two legs of the nuclear triad - nuclear gravity bombs/cruise missiles and the Navy's missile subs - were not secured in such a way. The bomber weapons were by necessity more vulnerable to potential theft (and, crucially, were intended to be handed over to allied powers such as Germany in the event of war), while the Navy's subs were so much less vulnerable that a short delay was not considered a critical flaw.
The notion that either the President OR the military has the ability to declare a strike without being checked is only true in older eras - PALS were ordered to be installed in the 60s under Kennedy, were completed by the late 70s for foreign-deployed weapons, and were universal by the mid 80s.