Yes. Being a hero who helps other people (not just a 'hero' in Skyrim who goes and retrieves the Lost Book of Fubar for some NPC who spends all day standing in their house) would be great.
SimCopter was a great one, you flew a rescue helicopter and would put out fires, chase criminals, pull people from sinking ships and take injured people to hospital. Heck, sometimes you'd need to carry the injured yourself. Fires and problems would be breaking out all over the city and you'd be the one helping to keep things under control. Very satisfying.
I think that might be what LA Noire was missing - you didn't really feel too much like you accomplishing something or doing much good. In a way, I suppose, that was part of the story (corrupt force) but I didn't come away feeling like I'd made much of a difference.
I already mentioned Crackdown, in Crackdown 2 the city functions somewhat normally during the day but is overrun by mutants at night. Sometimes you'd encounter a lone civilian out after dark being chased by mutants. Siren on, swoop to their rescue, smash the mutants and save the day. Felt great.