I see.
Of all the suggestions I've received down the years, I would consider Neon Genesis Evangelion to be the best one that I actually watched. Without trying to sound like a pretentious prick, I think it had a lot to say and managed to execute what it wanted to say excellently. I don't want to watch an anime about a cool guy fighting cool monsters with cool moves in cool places- shit means nothing to me, but Evangelion meant something.
The horrible situations, I think, added to this super deep portrayal of characters larger than a few traits. Yeah, a lot of the time they were played up through cartoonish traits, but ultimately the suffering brought out much more interesting situations.
I don't know if I'm the one to offer a breakdown of the things I like- I'm not exactly eloquent or well-versed. Suffice to say that I think there is intrigue and to an extent beauty in the nature of suffering, particularly that offered by Evangelion's brand of suffering, which has a way of leading to development and exploration of humanity's imperfections. Not one character in the show is really whole. I genuinely, profoundly love that.