It WASN'T his watch, he was never ordered to watch the Emperor. Could you imagine the emperor telling him "Oh hey Tulius, I know you're busy with the Stormcloaks, Dragons, and Dragonborn, but can you protect me personally too?..."
No, because Tulius' main objective is to win the war.
If the civilian leader of your entire empire, your boss, the Big Kahuna, the one who sent you there in the first place and from whom you derive authority, is in your territory (and here I'm not going to entertain the idea that military governor is different than any other sort of governor- he's the empire's man in Skyrim, they don't have anyone else in charge there) then it IS his responsibility. If the game were realistic, the Legion should stand down as of the empire's death, because there is no one on the throne any longer and hence no orders.
If such a huge fail can happen under Tullius' responsibility, that must mean he's a terrible general, right? Useless, to use your terminology.