It's pretty simple, really. The missiles require the simultaneous turning of two keys as part of the launch sequence (that part has been declassified for ages, which is why it appears in film so often), so as to ensure no one person can launch the missiles as a unilateral decision, preventing people from going rogue. If the plumber ambushed one of the key-turners before they could turn their key and smacked them over the head with a pipe wrench, now there is only one person in the launch room who wants to launch the missiles, so they can't do it because the keyholes are too far apart for one person to turn simultaneously on their own. Many of the silos had very small crews, and they were often lightly armed, if at all, relying on exterior security more than their own armament inside. The plumber wouldn't even need to keep others from entering the room all that long, just barricade the door long enough to keep others from entering until the incoming enemy missiles struck the site and made launch impossible.

You're right that many people wouldn't try and be a hero, but the fact that somebody could potentially try something when they had a chance could explain such a policy being in place. Many people surprise even themselves when it comes to intervening to save the lives of others. If you yourself were to be in a position where inaction means you survive, but millions of other people die, but taking action means there's a not insignificant chance you can save millions from death at great risk to yourself, that you are absolutely sure you wouldn't risk it? Especially given that in this scenario you're almost certainly going to die relatively soon anyways due to the aftermath of a nuclear exchange? If you have little to lose, and can save a whole shitton of lives, can you honestly say there isn't any part of you that would want to give it a go? In practice most people probably still wouldn't (bystander effect, basic fear, etc) but a non-zero number of people would, and many wouldn't even know if they would be the sort to do it until placed in that situation. The military wouldn't want a non-zero possibility of interference, so a strict contingency plan isn't so out of the question, IMO.