Oh, I can see how My Hero Academia would be an interesting first anime. It is also a perfect showcase of what you don't like about them.
I don't even remember the first anime I watched. It was most likely a part of the Japanese World Masterpiece Theatre from the period up until 1985, however these shows are also by far and large somewhat removed from usual anime aesthetics (well, usual for today, Heidi for instance all but began the kawaii (cute) looks of small girls in anime) and more grounded in terms of storylines because the had to adhere to children's book classics.
Other animes that I watched very early on were Attack No. 1, Bismarck (also known as Saber Rider and the Star Sheriffs), the anime adaptation of Captain Future and two interesting hybrids, namely Galaxy Rangers and Ulysses 31, which were written and produced in the US and France, but animated in Japan. Watching these shows nowadays it is painfully obvious where the actual writing comes from. Heh. Galaxy Rangers might look like 80ies anime, but it is so American.
All, or at least most of these, lack what you describe as problematic in terms of animation, artstyle and characterization (Saber Rider is even a "typical" space western) even though they are noticably Japanese in origin (except for the episodes of Bismarck that were made for the US market, in the US - those are actually painful to look at when you like the show). What I can clearly remember is vastly prefering shows that turned out to be anime (as if I had any idea when I was 6 or 7 where a cartoon on TV came from) over shows that turned out to be regular Western cartoons. With some exceptions - I'll always love Bugs Bunny or Tom and Jerry.
It should also be noted that I have seen a grand total of one (1) of the really popular animes that became world wide smash hits, and that, interestingly enough, was Sailor Moon, and yeah, back when I watched this was more of a niche underground hit than the phenomenon it exploded into. Which means I have seen none of the Dragonballs, no Pokemon, Digimon, One Piece, Naruto or whatever else might be the bee's knees these days.
I do need to point out that in terms of aesthetics I'm the polar opposite of you. I very much prefer anime artstyle and looks over their Western counterparts. It's not something I can argue beyond subjectivity, but as I said, it's always been that way. I'm with you in terms of writing and characterization, there's a whole lot of angst and overexaggerated drama going on at times. In an ideal world you end up with something like Castlevania, animated in Japan but written more sensibly than what could have been. I'm not sure why nobody else on the forum seems to have liked it. It was entertaining and looked good. Like BoJack's mother would have said: It's no Ibsen. But... what is?
And lastly, I can watch TV shows with great dedication as long as I find the characters interesting and I like them. Far beyond any reasonable point in time. I brought up Andromeda earlier, I loved that during its original run, and have spent many an hour defending its ludicrous inanities it reached before Season 5 finally broke the camel's back (well it's not the show's fault that Lexa Doig was pregnant at the time, but without her character there I stopped caring).
Which brings me to a favorite point of mine, one of the very few sitcoms where I've seen the entire run. No, it's not going to be about The Big Bang Theory, I watched that initially because my wife liked it, and while she actually stopped watching later after it became a full blown relationship sitcom with creepy characters I was coerced to finish it - by my brain. In hindsight I should have known better than to be roped into anything made by Chuck Lorre, but... eh... yeah. Anyway, back to How I Met Your Mother.
Generally I don't like sitcoms. I don't think they're funny most of the time, but I can appreciate certain novel ideas in one. How I Met Your Mother was one of them, featuring Ted as an unreliable narrator and copious amounts of flashbacks and flash-forwards that revealed a certain pre-planning that each season must have gone through (or the writers managed to come up with that by the seat of their pants, both would be almost equally impressive for a sitcom).
I watched the first two seasons and got a few chuckles out of them (I consider this a big win for sitcoms in general), so I kept watching, and at the end, I still liked it. Except for the ending, that was clearly because the show and its characters outgrew its intended ending and they just shoehorned it in anyway because the kids were all grown up already and they couldn't do reshoots. At this point I want to say that I also watched Lindsay Ellis' analysis of Game of Thrones where the exact same problem comes up, thanks for mentioning that earlier while we still spammed the TV topic :p.
Most people I know who also watched dropped the show way earlier. I can see where they're coming from. Ted is an idiot, and if you actually care about the storyline of him finding the mother, then it all becomes really ridiculous after the third season. But caring about Ted and his quest to find the love of his live is like caring for the central plot of any given Sailor Moon season. You can do that, of course, but that will only diminish your enjoyment of the show (well it's not like being interested in the plot is something you can actively turn off, at least I can't, but if something's plot is uninteresting but the characters are strong enough to carry the show, I don't mind that much).
The other big argument was: This stopped being funny in season <X>, often 4, 5 or 6, depending on whom you ask. No, it didn't. It never was funny in the first place, it was at best mildly amusing. As such, no longer being funny doesn't bother me either. I kept watching, and the writers eventually caught on too, I think. Some of these later episode were wonderfully written and directed (generally, not just for a sitcom), and some didn't even have attempted jokes in them (Symphony of Illumination is a masterpiece of an episode).
Except for Neil Patrick Harris, I don't even like any of the actors invovled in the show. I avoid movies with Jason Segel and Alyson Hannigan, Josh Randor is... err... I don't even know what he's doing, and Cobie Smulders, oh boy, I cringe almost every time I see Maria Hill on screen. Yet, they mostly worked in HIMYM. I liked Marshal, Lily and Barney, and even Ted to a point. I put this squarely on the amount of effort put into direction and writing for the show (or else everyone but NPH was playing themselves).
Wow that became long and rambling.
