Your idea of what "makes sense" completely obliterates the foundations of several characters moivations. Bane lets Gotham linger on for five months because A) that's how long it takes the core to lose stability and B) to him, it mirrors his experience in the Lazarus Pit where he could look up and see hope every day but know that he was trapped in despair. This is all very clearly expained and it ties together perfectly. They are breaking your logic (and mine) because they are, indeed, insane. But they are not breaking their own logic which is warped and driven by a need to make others feel what they suffered. To just obliterate Gotham instantly takes out a key element (hope) that Bane feels is necessary for true despair.
Also, when he says that an ordinary citizen has the detonator, I never took it to mean that they forced some guy at the stadium to take it willingly. I assumed they meant that a citizen who was empathetic to Bane's ideals was working with them (which is exactly the case) and Bane let them all know it to twist their perspective around and convince them that he's fighting for them, not against them.