I think Morrison's interpretation of the 'filthy city' cliche is exaggerated as hell, to put it mildly.
Rorschach and Batman see the dark underbelly of the city that the normal people don't see, and that's reflective of the real world -- excusing the fact that there are the occasional supervillains. When they say 'this city is filthy', there is no implication that there isn't a decently functioning, 'normal' society hovering ontop of that filth. And in fact, we see plenty of examples of regular people living regular lives all the time.
Another question posed is whether or not 'normal society' is really as healthy as it pretends to be. Crime isn't the only problem society faces. Sometimes the books are about poking at the holes in society's facade of stability or goodness. This is where some of the major villains step in to give them a little push.
The 'cesspool' in a literal sense is very rarely portrayed in Batman, from what I've read and watched throughout my life.