I think the reason I dislike the Joker being given backstory is not that I don't want to feel sympathy for the Joker (I mean, that's a small part of it) but rather that I think it undermines the point of the Joker. I don't mean as a character with motivations, I mean narratively. Others have probably said this before, and better than I'm about to, but this is my blog and I'm going to ramble.
The whole point of the Joker that some writers seem to miss (especially films! films are ADDICTED to backstory for some reason) is that there IS no real reason. He's a primary antagonist because his entire schtick is that nothing matters, and that's funny as hell. He can treat people's lives as a joke because they don't mean anything.
Bruce, or Batman, as the protagonist is the exact opposite of this. The motivation to save others from what happened to him as a child is rooted in this concrete, fundamental-to-Batman truth: everything matters. This is why Bruce's parents dying in a random act of violence in a before-then obscure alley has had such staying power: it wasn't a massive event that changed the lives of dozens or hundreds. It was a small thing, a statistic in the grand scheme of violent crime, with no real motive other than desperation or greed. And it changed someone's life completely. For Batman, the hundreds of small acts that fill a patrol all matter because each person does.
Joker's obsession with Batman hinges on him believing the exact opposite, and trying to convince him that he's wrong, and that THE joke is that nothing matters at all-- walking away from a victim and torturing a victim have the same moral weight because there is no moral weight. The point isn't to be the most evil or the most cruel, the point is to prove that it's all pointless.
In a way, this means Joker's greatest ally is the apathy of Gotham. Every person who turns away or keeps walking with their head down and claims "not my problem" supports his claim that nothing matters, because Gotham is proving by inaction that it's pointless unless it impacts them personally. Nobody else matters unless I do, and then it's too late. Do I think Joker's crimes are Gotham's fault, narratively and within their fictional world? Not exactly. But I do think Joker isn't actually an embodiment of chaos but is a representative of what Batman is actually fighting against: a Gotham that doesn't care.
Batman's existence is a defiance of this. His work, masked and unrewarded, is a symbol of the fact that no wealth, no material comfort, no border of security, no past trauma, no law, no excuse justifies not caring. Every life matters, even if it isn't your own, even if it's a stranger. He can't kill Joker in-world because of his moral stance on killing. He can't kill Joker narratively because Gotham won't kill Joker. Batman's driving core is a relentless anti-apathy, and Gotham's current collective social spirit is that they are helpless and not responsible for it.
Joker thinks it's funny that anyone would care, and Batman thinks someone not caring is the opposite of funny. Caring can often hurt, but it isn't pointless. Every single life saved or lost creates a ripple effect that matters and hurts or helps others. Each victim Joker encounters is met with the same presupposition: "I can hurt you and it doesn't mean anything because you're nobody. It doesn't hurt Gotham because Gotham isn't anyone."
Every person Batman saves is saved by the same presupposition: "If I can save you, it's like saving all of Gotham. Everyone matters...because each single life matters."
The more people that build their lives on caring regardless of how they think it impacts them personally, the less sway the Joker has. Every person who shuns apathy is another person who will fight back against the Joker's Chief Joke: It doesn't matter.
And they fight back best by, like Batman, looking at each individual they come in contact with and deciding, "Yes, this one matters."
