Here is my headcanon why Batman can't kill criminals.
We all know the personal reasons why he doesn't: Doesn't like guns, believes life is precious, trauma could make him used to killing and turn him into a villain, etc.
But to justify why he doesn't let others kill, is worth looking at the general context.
It's quite simple: Gotham adapts.
Keep in mind that the current era of masked criminals is in part a consequence from The Long Halloween. Before the time of freaks and gangs, there was the era of mob bosses and families.
Who was The Holiday Killer, the one that finished the era? Either Gloria Dent or Alberto Falcone, victims directly or indirectly affected by the current status quo, who took letal justice for their personal reasons.
In a way, Red Hood is recreating this scenario. Similarly he is a victim of these current evil forces who wants to take letal justice against them.
You could argue that Holiday did the right thing, killing the people hurting Gotham and changing how things were run. But did Gotham become safer? More peaceful? Not really.
The violence and murder still exists, even without the families running Gotham. But in a new form. The horrors are there now hidden under masks.
If Batman, or Red Hood, started systematically killing the big criminals in gotham, the process would repeat.
Do you really think that the criminals in Gotham wouldn't react if big villains were being killed?
Chances are, they would be killed after the initial killings. because getting to them is way more difficult than with past normal criminals. Remember, some of the Rogues have taken over the city. Do you really want them to unite against one enemy?
A VERY talented vigilante could probably get away with killing one or two villains (Probably the Joker, honestly). But finishing the job? That would require a lot of talent and miracles. A true massacre, with the city as collateral.
And if they miracly finish the job? That wouldn't mean peace. We already seen various villains follow on the step of others. A retired Riddler lead to other three trying to take his place. Punchline is following Joker's steps. There's probably more villains that would rise up after such a massacre.
Just look at Batman beyond, Bruce's enemies might be dead, but Neo Gotham is still full of villains.
Batman's methods current aren't perfect, but it avoids a larger scale of crime. He slowly treats the symptoms, rather than risk bigger damage while trying to forcibly fix the city's issues.
The Rogues are already ten times worse than the Families were. How bad would the things that raise up be?
