Your criticisms towards Batman are a bit unfair IMO. Bruce Wayne always does something with his money to help Gotham (I'm talking about Bruce Wayne/Batman in general here, not just the Nolan version). Batman himself is responsible for taking out the trash. While Batman himself most certainly can inspire the people of Gotham in both good ways (i.e. Harvey) and bad (i.e. Joker), the main thing he specializes in is striking fear into the hearts of bad people. I forgot who said this, but I remember a DC writer saying once that "Superman is the force that guides society in the light and Batman is the force that takes out the trash in the dark" or something among those lines.
Well, yes. But what makes the trash? If you are going to really talk about cause and effect (something that is brought up for Joker storylines or maybe The Long Halloween, but by and large is ignored in comicdom), everyday "trash" tends to fluctuate on the state of the city.
Even at its most basic level, why is Metropolis, which is drawn to be so bright, shiny and futuristic, free of most everyday crime in comic depictions while Gotham, which is drawn to be dark, dirty and malnourished in civic duty by a corrupt police force controlled by the mob, seen as overrun with crime?
Both acknowledge there is more to the problem than just hoods on the street. Nolan attempted to put Batman somewhat in a proximity where those questions could be asked and make Batman answer them in a way that was still heroic, just with an eye on a bigger picture. You don't have to ground your story like that, but if you do, I appreciate doing so in such a thoughtful way.
It is the difference between Nolan's approach and Snyder's in MOS. Snyder went after that somber, mythic tone with a grounded "realism" that Nolan captured, but he didn't really pay attention to the details of what the consequences would be for his character. Okay, Superman's existence apparently means the entire destruction of downtown Metropolis in free-for-all. How do people react? How does Superman react to all the death and destruction? How does the city respond to losing thousands of people? It doesn't, because Snyder goes back to the status quo of Clark Kent at the Daily Planet without an explanation or a reaction to all the devastation beyond a make out session.
I know this is rambling, but the astute attention to these details is why Nolan's Batman can seem so authentic in his context and MOS can seem so awkward in its grim, but very cartoonish setting. If that makes sense.
