I am not asking "why doesn't Batman kill?", he has his own moral code that he abides by to avoid becoming a villain himself. However, Gotham City should have no such qualms. It would most likely be extremely popular with the general public, and America still practices execution.
Are you telling me that after however many trials, arrests, and breakouts, they still put Joker in the asylum? Obviously, out of universe, Batman's "rogues gallery" is important to the story; but what about in-universe explanations?