They are chock full of the worst tropes in anime. edgy grimdark, random death and violence. if that's your thing by all means go for it, but the writing is terrible, the voice acting even worse. I highly recommend to anyone who is visiting this thread who hasn't watched anime yet, to NOT watch these two shows. You get what you deserve, the source material is literally teen trash, erotic novels marketed towards young adults. It's like expecting a good science fiction movie to be good thats based off of a poorly written preteen science fiction novel. -cough- hunger games and maze runner -cough-
I'm not sure what you're talking about regarding the tropes; when I think of "edgy" shows that use random violence, I'm more likely to think of Fist of the North Star or a lot of the stuff that Manga Video brought to the west back in the 90s. Also, while Akira is a great film, one of it's primary reasons for success in the west was precisely that it had extreme violence which was unusual in "cartoons" at the time in the western world (certainly wasn't the story, as I guarantee that few people understood Akira's story the first time around, and some people still don't), since cartoons were typically always targeted at children in the west.
Most people's negative perceptions of anime typically come from either associating it with childishness because cartoons are usually for youths, or an association with "weeb culture" and moe archetypes, etc, which are viewed as cringe-inducing by non-anime fans (and even many anime fans too). In fact, if I had to think of the "worst tropes" in anime, the ceaseless moe spam would have to be at the top of the list, as the "edgy violence" tropes you point to aren't even common enough in modern shows for it to be an actual problem.
As to the criticisms themselves, maybe you were watching different shows, but the violence and deaths aren't any more "random" in those shows than they are in a show like Game of Thrones. I should know because I watched a massive amount of "random violence" anime as a kid because violent edgy stuff is the only things they used to bring stateside back in the 90s or so, so it was all I could find at local video rental stores. Shows like "Angel Cop", "Wicked City", "Genocyber", "Fist of the North Star", "Devilman", and even more popular ones like "Ninja Scroll" would fit your description of random violence for the sake of edgy gore far more than Higurashi or Madoka, which have actual stories that captivate. As for the source material, Higurashi's source material was a detective murder/mystery game series, where the goal is to collect clues to solve murders, and the show has a similar feel to it, where you never know whether the characters or a scene in question are real, or if it's some hallucination, and knowing the difference is typically the key to knowing what is actually happening. Madoka meanwhile doesn't have "source material", as the anime is an original production that
then spun off manga and novel adaptations, not the inverse.
Also, it's odd to me that you added Hellsing on your list (albeit with a "once you're comfortable with the medium" disclaimer), which while it's a good show, is
far more guilty of peddling random violence for the sake of violence and being edgy. Much of the plot is simply an excuse to watch Alucard go nuts and kill everything in sight (not that it's a bad thing to want to see that, after all, horror movies wouldn't be popular either if there weren't something fascinating about violence to a certain audience).
Now I will give you one point, which is that in this specific community, with some people who may be more middle aged, and who may have more of a lean towards the creative/imaginative aspect of art, perhaps some of these shows would be boring as this demographic may be "over" the whole ultraviolence thing. But once again, this would equally apply to shows like Hellsing which you mentioned.
And while I hate to pull the "popularity" card, while Higurashi is a bit more niche, Madoka certainly is not, and is a massively successful franchise. Popularity does not make it good, but it raises questions, there is no such thing as an "objectively good" or "objectively bad" show, it's all subjective, but it's certainly weird when someone says "all the Marvel films suck", or "all of Game of Thrones sucks"; again, popularity doesn't make these good, but it does have
some meaning, namely, that the shows in question have some mainstream appeal, and thus maybe it's not crazy to recommend them while thinking that someone else will enjoy them too.
Lastly, while Cowboy Bebop isn't bad, it's not for everyone, the episodic format can be very frustrating for some people. When I first watched it, the first episode was really exciting, but then as it went on and I realized there wasn't really an ongoing story, and it was more just watching them go about their day, etc, I just found it hard to get invested in it or interested in what happened to the characters, I felt like I was watching strangers for the whole show, as if each episode is episode one. The music is great however, which has to be noted since this is an audio forum. =)
Oh, and FLCL is one of my favorite shows of all-time as well, but I don't know if I would call it "approachable" for people who are completely new to anime, and have a bad impression of it, it's pretty
out there, and I think most people new to anime as a medium would be turned off by not knowing what in the hell is going on half the time (half is being generous here, more than likely, most people won't know what is going on at all from start to finish).