That same logic applies to everyone else too. If Batman has some obligation to play judge, jury and executioner, so does anyone else who gets the opportunity. By that reasoning, cops who see criminals getting back on the street again and again should take matters into their own hands and just kill suspects who they believe are too dangerous to entrust to a broken system.
Batman isn’t just trying to beat up every criminal in Gotham, night after night, one by one. He’s trying to clean up the city, take down organized crime, fight corruption, and help fix the system so that it can deal with crime. Killing people prevents him from being able to work with people like Jim Gordon. And it forces the authorities to redirect resources away from fighting other criminals and instead use them to hunt down Batman because he isn’t just a vigilante trying to help, he’s a mass murderer.
And while that may prevent his victims from coming back to commit new crimes, it won’t prevent new criminals from taking their place, something that’s going to happen when he creates a power vacuum by killing a major crime boss. Plus, the calculus changes for any criminal he faces, because they have far less to lose by doing something stupid or desperate when their lives are already on the line. It destabilizes every situation, and makes dangerous people even more desperate.
Any lasting solution for Gotham has to involve fixing its failed institutions. That’s a long term goal, and one which Batman can’t do by himself. He can make a difference, he can protect people on a daily basis, and fight corruption wherever he’s able to find it. And he can inspire others to act, coordinate with them, and support those who try to create a real change.
Now, will he ever actually clean up Gotham? No, because then we wouldn’t have a story. Of course, that would be just as true if he killed people. I mean, Frank Castle has been gunning down criminals for decades, and yet, that doesn’t seem to have eliminated violent crime either.