Disagree with everything you said.
I mean, I think the question is answered in the original post. The Burton films inspired the Animated Series and much of the visuals. Batman the Animated Series had a few campy elements, but its underlying themes were incredibly, even shockingly, adult and psychological. Im always surprised by just how well they hold up by the standards of today. The show isnt as gratuitously dark as the Burton films, but in spirit it oftentimes was. Dini and Timm just had to find clever ways around the kids cartoon show misnomer, because its clearly not intended for kids. So its more subtle compared to Tim Burtons live-action films, but the darkness is still there, you just have to look for it. Its not blatantly apparent.
Take the episode Jokers Favor, where a man cuts the Joker off on the highway and Mr. J. just isnt having that. So he tails him, tracks him to this back woods, and threatens to kill him and eventually his family if he doesnt do him some sort of favor. Nothing is ever firsthand said, no threat is outright mentioned, but its clear as crystal. It gets so obvious that the man actually moves states away and changes his name yet is somehow still tracked down and phoned by the Joker himself, who tells him its time for his debt to be paid. He then curiously looks out his window, which you see his little boy playing with a foot ball and wife sunbathing
and Jokers goons drive by slowly and suspiciously. Oftentimes what was forced to happen was that the Joker or any other villain attempts a killing and is heroically stopped in the nick of time, not that the violence is inherently not there, but rather its sometimes gruesomely attempted, just not carried out to its end due to the censors. Oftentimes the attempt of violence is every bit as grim as seeing it carried out.
The other thing is that Batman the Animated Series was somewhat novel at the time for its use of real guns. In the early 90s, parents would group together and form childrens safety boards, *****ing and screaming at any cartoon that depicted even mildly realistic violence, fearing wed become desensitized and blow our classmates away. Its really astonishing how Bruce Timm got away with what he did, somehow probably only barely passing the censors on most episodes. Consider the Marvel cartoons of the 90s, when authorities had cheesy laser guns instead of pistols and rifles. Look at G.I. Joe: Real American Hero and Cobras infamous terroristic light bright display of weaponry, their only actual threatening arsenal, missiles, always comically missing their target by miles. Not only did the Batman series manage to pull off actual firearms, but time period accurate as well, like the Tommy gun, which was another thing that set the show apart.
Its set in this Tim Burton inspired film noir world. I completely disagree that its not. Look at the emphasis upon shadows and darkness covered caricatures in the very opening credits alone! Not to mention both the show and Burtons emphasis upon Pulp stylized gangsters with their Dick Tracy-like zoot suits. I dont mean to sound rude, but do you know what film/gothic noir even is? The Animated Series is some lost time period with contemporary, even sometimes futuristic by todays standards, technology exists alongside black and white televisions and 1930s and 40s architecture and automobiles. You cannot directly associate this with any pinpoint era; it both blending this timeless classic pulp detective drama with modern Dark Knight Tales, really added some unparalleled artistry to the entire show.
Christopher Nolan doesnt have much art in his soul when it comes to eye candy. For all his talent on the page, hes lacking in visuals, possibly due to his real-world agenda, whereas Tim Burton is arguably vice versa. Tims almost overly-obsessed with the imagery itself that he prioritizes it over the story. Example; Nolan Gotham City is merely background; its a character in Tim Burton and Bruce Timms worlds. Which is one of the many reasons why I think Batman the Animated Series is clearly inspired by, something that is an admitted fact, and toe-to-toe with the Burton films, merely toned down.