Basically, the US president is in the constant implicit position of being able to hold the entire US hostage from a flying doom fortress† like some kind of supervillain. The only people who can really stop a “crazy president” scenario are people either high up in the Secret Service or the Joint Chiefs of Staff, who would 1. Have the authorization to be in the president’s presence during DEFCON 1, and then 2. Have the personal trust/authority to tell the president’s bodyguards to buzz off for a minute.
† The “doom fortress” part applies more to the subs than to AF1, but if you treat the two as a unit, it works.