There wasn't a backlash about it because the times have changed. Some people let their children watch shows like True Blood and things like that these days. Hell, I remember some overly religious types taking their 6 year old children to see Passion of the Christ when it came out, even though it was more gory than any horror movie I've ever seen. I also don't remember anyone objecting to Kate Winslet's nude body being shown in a PG-13 movie, and that was over ten years ago.Showing a man screaming in agony as his body burns (immediately after his girlfriend is murdered, no less) and then showing the after effects isn't violent? Showing a bunch of thugs shooting each other to death and running one over with a bus? A man getting a pencil impaled through his eye? Car bombs, drinking acid, three corpses with grins carved into their faces, a man being sat on fire while he sits atop a pile of money? Citizens trying to assassinate Reese due to Joker's threats against their loved ones? Batman breaking Maroni's leg? Two-Face holding a gun to a child's head? None of this is violent?
If anything, it was far more extreme than what happened in Burton's films. His films had Joker shooting people, gassing them, and an extremely cartoony joybuzzer scene. The second one had Catwoman scratching someone, stabbing Batman with her claws, Penguin biting a man's nose, Batman setting a guy on fire, and him strapping a bomb to a man's chest. While these acts were no doubt violent, they still were more often than not portrayed in a cartoony manner, and weren't nearly as extreme as what TDK showed, IMO.
Also, from what I remember, the protests against BR were largely due to sexuality. Penguin made several sexual remarks towards Catwoman. Catwoman made various references to Batman's anatomy and dressed in an S&M costume (complete with a whip!) and licked him while sitting on top of him. TDK didn't have anything remotely sexual, from what I recall.