Klasky Csupo's style worked very well in the right contexts. I think Rugrats was one of those contexts, because the whole world was supposed to look kind of weird, because it's through the eyes of babies, and everything is weird for them. Also they themselves are pretty weird and gross when you observe them, and that's part of the show.
The only times I really think their style looks weird is As Told By Ginger, since that show tries to be slice of life realistic stuff. A more realistic style might have worked better. Then again I don't like that show anyway. On the other hand, Ahh! Real Monsters! is a perfect fit for the style, because it's monsters, it's supposed to look weird.
I think they're all supposed to have flaws, but Angelica's parents are the only ones that are really bad. Stu is too childish and sometimes that results in irresponsibility, but it also results in him having a great bond with his kid. Didi is very well intentioned but represents all the problems with parents who try to parent from a book, listening to academics who are ultimately just talking shit. Chaz is obviously loving and protective, but his extreme protectiveness results in his kid being a scared little nerd. Phil and Lil's parents get less focus but it seems like maybe they're too lax and thus their kids are little weirdos. On the other hand, you have Angelica's parents, they work too much and don't spend enough time with the kid, but when they do, they spoil her with too much undeserved praise (perhaps because they don't spend enough time to build a real relationship and treat her with more nuance). Then when they do lose it and finally punish her, it's over the top, and sometimes she doesn't even deserve it. There are a lot of episodes of Rugrats where Angelica was being bad, but then she gets punished for something completely unrelated, or something she actually didn't do. Or times when she really didn't do anything but gets punished anyway, because the viewers understand she's a little cunt and deserves it either way. But it makes sense for the parents' characters regardless.
Rugrats is deep. But every season after 3 gets progressively worse, and the fact that they've managed to make the new stuff even worse than the 2000s-era seasons is almost impressive, because those 2000s seasons were already awful.