I don't disagree with your analysis and it is what I think they were going for but at the same time it doesn't make much sense in "reality," Gotham still, very much, had countless problems beyond what the local police department(s) and justice departments could do.
But the point isn't that all their problems were solved -- the point is that the people themselves had become inspired to stand up and address those problems themselves. As was symbolized when the people of the city (well, mainly Tiny Lister) refused to sink to the level the Joker tried to drag them down to, and essentially saved themselves.
I hesitate to invoke the "teach a man to fish" cliche, but it kind of works here. Nolan's Batman is more about being a symbol to inspire others than about doing all the do-gooding himself.