‘Batman should kill the Joker’ discourse never contends with canon datapoints like that time the Joker thought he was dying of cancer so he Jokerfied the pretty much the entire planet and was going to be replaced by a legacy Joker who also had superpowers, or the fact that the Joker has to deal with people who want to kill him so that they can become the Joker, or that time that everyone thought Batman killed the Joker and so a new guy decided to be the Joker until the old one got PO’d at him and came back
There's also takes like Batman: The Audio Adventures where it's outright said that one of the biggest things protecting Joker is the fact that he has a whole widespread cult of personality who would be causing chaos and/or killing themselves and the only reason they're not is because the Joker is around to tell them not to. Which I do think you can sort of surmise is also the case in mainline canon at this point even if it isn't said quite so succinctly.
and its important that this is a given--that killing the joker wont work, or will be a net loss--because you are being asked to think around it. you are being asked to come up with a different solution. writers will come up with a million different reasons why batman can't kill the joker, and none of them are as important as the question:
with execution off the table, what else can you do to solve this problem?
I think the object thing here is what Joker represents. As with everything in comics it’s not always consistent, but it tends to wrap back not just to chaos as a general concept but the specific idea that you could be the victim of sudden, random violence at the hands of a stranger. Someone could push you onto the subway tracks in front of an oncoming train, they could knife you as you’re walking home, they could pull out a gun and start firing indiscriminately into the crowd. These are inherent risks of living in the world that can’t be solved by simply killing the ‘right’ people—there will always be somebody else out there who could hurt you, even if they haven’t decided to do so yet. There will always be another Joker.
This is also why it’s significant that Joker doesn’t have a civilian identity or stable backstory. Broadly speaking, the way that DC characters—and Batman characters in particular—work is that they’re divided into a costumed identity, which represents big, thematic ideas, and a civilian identity, which explains their motivation for engaging with that theme, often via traumatic backstory. Eg., Batman is a symbol of vengeance and justice because Bruce Wayne is haunted by the night he watched his parents get murdered as a helpless child. The fact that Joker has no civilian identity or backstory points to his apparent lack of motivation for hurting people, which is to say, it’s senselessness and unpredictability. He’s the perfect villain for someone whose life was diverted by an arbitrary act of violence.
