Short form: it's terribly easy to get the wrong idea about anime, even if you've watched it before, casually, on American TV. Because even if you know something about Japan and its culture, some anime can be...inaccessible.
I like to bemoan the fact that three basic types of anime are all that makes it to American TV: grim and gritty (Cowboy Bebop) or super-long-running kids series (Pokemon, Dragon Ball) or shounen series (Death Note, Full Metal Panic/Fumoffu).
I don't watch any of that stuff.
We used to get magical girl stuff too (Sailor Moon, Cardcaptor Sakura) but I haven't seen any magical girl anime listed since the late 1990s. And that was Cardcaptor Sakura in its completely hacked-up, censored, and sanitized form ("Cardcaptors") because, you know, we can't possibly show the 11-year-old girl with a crush on an older boy being humored by the older boy or anything. 9_9
(I did watch Gundam W when it was being run on Cartoon Network, and I collected the DVDs. "Giant mech" stuff is also easy to digest, but it falls under the "shounen" category above anyway.)
There's a reason this stuff is brought to American TV: it's really easy to understand and fits the American taste rather nicely. Unfortunately, it's not even close to a representative sampling of anime.
Probably 90% of the stuff that I like these days is "slice of life". It's not about anything big and super and magical; it's about more-or-less ordinary people dealing with more-or-less ordinary situations. Kimi ni Todoke is an example; it's kind of on the border of romance anime, too--let's face it: it's a shoujo series. Little-to-none of it would do well on American TV; the target audience would denounce it as "boring".
The "target audience" being, of course, ages 12-22. (Younger for Pokemon and other juvenile anime.) Of the three categories I listed above, only the kids' shows are aimed at a younger demographic.
Besides, how the hell do you market Marmalade Boy or Kimi ni Todoke merchandise? Any viewer old enough to appreciate the stories is beyond the age for toys, and there are no video games of this stuff. It's just not possible to make any that would be a commercial success, particularly since even now anime is still a niche market in the US. (It's a lot bigger than it was when I got into it, in 1994--a hell of a lot bigger!--but it's still a niche.) A certain small subset of anime viewers would shell out money for figures and maybe themed merchandise (notebooks etc) but not enough to make the expense worthwhile.
Meanwhile, go to the nearest big box store and browse the toy section for Pokemon crap. That show's been in release here since 1998 as I recall; it's been twelve years and you still can buy all kinds of merchandise for it.
The point is, the stuff on TV has already been pre-filtered by people who want to make a lot of money on commercials by showing it; they're going to pick stuff which looks like it'll go over well with the American public.
Stuff that's not likely to do well never even enters consideration. Something slow-paced like Sketchbook: Full Colors would be rejected in the first round of brainstorming. Kanon would never be shown; too weepy in the middle and there isn't any real external conflict. Something like Arakawa Under the Bridge would just be too weird.
In actuality, I think the success of the Canadian Degrassi series tends to show how well "relationship" anime would do--KnT, for example, or Marmalade Boy--if it was presented correctly. Problem is, "correctly" involves doing more than just translating and dubbing the thing; it involves marketing, which takes money no one wants to spend.
Look: people are people, no matter where they come from; there's no such thing as a "unique Japanese situation" and the emotions of the characters are universal. Even if you don't understand why a character feels the way he does, you understand what he feels. (If the anime is done correctly. Sometimes it's not.)
The issue of "accessibility" doesn't come from anything unique to anime or Japanese culture. It comes, in fact, from the fact that there is a lot of it out there.
If you watch something like Minami-ke as the first anime series you watch, you might conclude, "WTF is this? It's just about the lives of three girls living in an apartment? This is stupid! Nothing happens!" Well, that's exactly what the series is about; it's not about wacky cartoon hijinks or anything. It's a humor series but it's not like Family Guy or something; there's no element of the fantastic in it.
Someone I know was at a game convention and the only place there were any damn chairs was in an anime viewing room. They were running a Dragon Ball marathon; it ruined his impression of anime. (I can't say I blame him. I can't stand DB in any of its iterations.) Though he hates anime in general, I was able to show him some stuff that he quite liked. Nonetheless, the damage was done.
As an otaku, I want to
It's not because I think the stuff's too "enlightened" or that these people are too stupid to figure out what's happening. I mean, this stuff is produced for the common denominator in Japan, which (despite its better schools) is not higher than the common denominator in the United States. Sure there are cultural things which an American might not get, but unless it's a series which hinges on obscure points of Japanese culture that doesn't matter. (Haven't seen one like that yet.) Anyone can understand and enjoy anime.
It's because there are so many genres. It's because there's so many different things. In my career as otaku, I've been primarily into magical girl and slice-of-life anime, with some minor forays into fantasy and SF anime...and I'm just wading. I don't know anything about some major series (Full Metal Alchemist etc, for example). I've got hundreds of tapes--literally--and a crapton of DVDs and 500 GB worth of AVIs...and I have a small anime collection.
I can usually give someone who wants to try anime a handful of things they could try watching, but the thing I always try to stress is a simple rule: If you don't like the first thing you watch, try another. Hell: though I liked Speed Racer at age 5 and Star Blazers at age 12, when Sailor V first brought over Urusei Yatsura in the summer of 1994, it just pissed me off. Then he brought over Ranma 1/2 and I laughed my ass off, and that was all for me.
The important point: Later, I tried UY again, and watched as much of it as I could get my hands on. It ended up being a lot: about 3/4 of the 200 TV episodes. I like it now. The point is, if UY had been all I tried, I wouldn't be an anime fan now.
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Notes and miscellany:
The first season of Ranma 1/2 has one of the dippiest opening themes I've heard. I'm used to it now; but it almost soured me on that series.
Urusei Yatsura was probably the beginning of the tsundere schtick as it's now used in anime; it was one of the reasons I didn't like the show. Lum was always frying Ataru. Ataru is a jerk--everyone in the series is a jerk, except Lum--but I thought Lum was being unfair; and in fact that's what I usually think about most any tsundere: does the guy really deserve to be pounded into the pavement?
One of the reasons Japan scores so highly in the international rankings comes from the way their school system is organized. Compulsory education ends after the third year of junior high school; you don't have to go to school after you're 15 if you don't want to. Students compete for slots in schools via tests; and of course asian cultures in general have a very strong education ethic. So the combination of competition plus allowing slackers to quit raises the score.
(I think the US should adopt an education system similar to Japan's. But of course the teachers' union wouldn't like that.)
Whenever anyone makes noises about wanting to try anime, I always want to quote Rassilon from the classic Doctor Who episode, "The Five Doctors". There's a scene near the end where Borusa is about to obtain immortality, and the image of Rassilon, with a grim smile, says in a sinister tone, "Be sure! Be very sure!" I always want to say that. Instead I quote Otaku no Video: "Once you stay here, you can never go back."
(Pretty much, whenever you're making a choice and the person who'll grant your wish tells you, "Be sure! Be very sure!" in a sinister-yet-jovial tone, RUN AWAY. Whatever it is, it's not worth the complete fucked-up-ness that will become of you.)
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I make light of otaku-dom and pretend it's some kind of sinister disease or something, but it really is a great deal of fun. Certainly it's no worse than SF/Fantasy fandom.