Oh sure, it is a good idea to take responsibility for ones actions, but there is a difference between taking responsibility for your actions and considering possession of power to inherently carry an equal quantity of responsibility to use that power on the behalf of others.
If a person has power that they do not want and makes the decision to not use that power and accept the consequences of doing so, there is nothing inherently wrong with that. Suggesting that they have somehow failed by not bearing an equal amount of 'responsibility' to their power is not an ethically sound argument. Being able to do a thing does not mean you must do the thing, only that you can choose to do so. The importance of choice is the entire point of the illusion of free will, and abrogating that choice should not be considered a good thing, even when justified by the 'greater good'.
Otherwise, where do you draw the line? Follow that logic and it's a surprisingly short path to justifying things like forced conscription. Being stronger and faster than the average person does not magically convey a responsibility to use that strength and speed to fight crime onto a civilian like Peter Parker, not unless he chooses to accept that responsibility for himself. (Preferably by joining the police, but apparently putting on a costume and being a vigilante became ethically acceptable at some point, so that's okay too I guess.)