No, not “responsibility”. What comes with great power is great “authority”, and that does not create greater responsibility, not necessarily. It would only create great responsibility if the one with power is a moral being. Without morality, there would be no responsibility: If Spider Man was to be a psychopath, he would never have felt any reason to go out in a dark, cold night to fight with a thief in a dark alley.
This is the reason why we should care whom we elect, and where we do that. Giving authority to those who do not care, will never create the best possible outcome.
And more importantly: This is one good reason not to choose any religious person to a place of authority, especially when they are extremists: What they do not have is the mind of reason, which is the only possible guide to being “good”.