My own preference is still "Less serial killer, more prince of crime". We have plenty of Serial Killer Joker, it gets old. Go for a version of the Joker who has grand aspirations like Bruce does, only his aspiration is to rule the city as it is, rather than transform the city into what it could be. He's an obsessive genius prodigy who manipulates, lies, intimidates, and murders in order to place himself at the top of the pyramid. And once he's there, what is the Joker's goal? Nothing. He seeks power because he can, arbitrary and capricious, and would exploit the power he has in equally arbitrary and capricious manners. There is no grand ideological reason why he would pick a random person and sadistically disassemble their life, not even simple vengeance; he would do it because its fun, and because he can, and because no one can stop him.
First time posting in here, although I've followed the thread for a while.
I agree with this for the most part. It's true that some of the most well regarded and iconic Joker stories are somewhat portrayed Joker as a serial killer if you reduced them to the basic details. Joker of course is introduced in Batman #1 as essentially a serial murderer. And stories like The Joker's Five Way Revenge and The Laughing Fish and Sign of the Joker are considered influential because they get back to that idea. Yet it's not as simple as that.
In Batman #1, while in both Joker stories in that issue the basic thing is that he goes on a killing spree, that's not the only aspect to it. The reason that this is such a strong introduction to the character is not just because he's a killer. If that was the case they wouldn't have decided to keep the character, rather than kill him off. What makes it compelling is that you get a sense that the character is cunning and audacious, which you can't help but enjoy. He is also rather ego maniacal and while he enjoys confounding the police and authorities, he also seems to desire more of a challenge. Which is why later he'll enjoy having the Batman as his best opponent.
The second Joker appearance in Batman #1, feels a bit of a retread as he escapes and resumes the killing and jewel theft spree, so this feels less strong than the first story which may be why it's not always collected with it. Imo the two Joker stories in Batman #1 are a two part story though, so should be read together. Perhaps that shows that just having him kill is something where less is more, but it also brings out the more maniacal, crazed aspect of the character. Specifically the first panels where he's in jail and of course the hysterical laughter at the end at the prospect of his own death. Which is something that some state only developed later as in the first Joker story he seems more stoic, but I don't really agree with that. The concept of him as a cunning, ambitious but maniacal and ego driven master criminal is there from the start really. Of course, Joker kills in other early 1940s stories too like The Cross Country Crimes and Laugh Town Laugh, but not always as they developed some other aspects to the character e.g. crime boss, themed crimes, humour etc. As you say, he mostly does these things because he can for seditious amusement and ego, but a grand philosophical cause really is not necessary.
Then when you consider stories like The Joker's Five Way Revenge and The Laughing Fish/Sign of the Joker, those are well written and well illustrated. But they also stand out because they provided a change of pace. Namely, The Joker's Five Way Revenge stands out not because the Joker is killing just because that's what he does, but because it was the first story in which he'd be shown to do that in a long time. Similarly, The Laughing Fish/Sign of the Joker stands out because although by 1978 The Joker had been depicted as a killer again, this was the first time that we get the motif of announcing kills before he does it for a long time.
So, basically, while stories like the above give the impression that killing sprees are a character fundamental, it's not as simple as that. In all of the above and others, it's the novelty and also well written scripts which offer other characteristics that make them great. "Kill kill kill" is not the only thing going on with the character, even in stories in which he does. That's why I find some newer Joker stories a bit tiresome as the kills seem there "just because" and lacking in suspense and impact because too much of this makes you not care. Joker is a murderer, but that's not all there is to him.