I was rewatching Azumanga Daioh a few weeks ago, because I hadn't in 3 or 4 years. It's still one of my favorite series, even though a decent chunk of its jokes don't land for me. Some of them are a lack of understanding on my part, the joke being a cultural reference I don't get. Some of them have a cadence that doesn't work. There are frequent jokes where the punchline seems to be an extremely drawn out silence, or someone (usually Osaka) repeating a phrase they'd just said while another character looks on incredulously. Which seems like it could be funny, but it usually doesn't work. Maybe it's a culture gap, too. There were also a couple of bits I realized probably wouldn't work now, like where they debate what a panda looks like, and they don't have smartphones to look this up on, since I think the time when this series was taking place is roughly contemporary with my years in high school. I'm going to try not to think too hard about that.
I read a description of the series once on TV Tropes that it tends to focus too heavily on Sakaki. Which is true in the sense that she does get a lot of time front and center. I tend to like her character, so I don't mind that, so I can't really say if she dominates the series. I think Sakaki gets the majority of the long-running plot threads (Tomo and Osaka are around a lot, but more in the brief joke scenes), with her love of cute animals, which runs smack up against the fact almost all cats seem to hate her. Also her own apparent discomfort with herself. She's very focused on the fact she's not "cute", and this is something she very much wants to be. Everyone else in the school thinks she's a really cool "lone wolf", but she's mostly just shy and awkward. I'm not terribly surprised she got more screen time than some of the other characters, given that.
There's also what I'd call a fair amount of fan service involving her, since she's a bit more developed than most of her classmates. Which is especially awkward, considering how uncomfortable she's presented as being every time someone makes a comment or reference to the size of her chest. She isn't even particularly comfortable when it's pointed out how tall she is, probably because it makes her feel even farther away from her conception of cute (which child genius Chiyo pretty much exemplifies). Then you throw Mr. Kimura in there, and that makes it even more awkward to watch.
Kimura is that kind of stock, "creepy older guy" character that a fair number of animes have. The first time we see him, one of his students asked why he became a teacher, and he loudly proclaims, 'BECAUSE I LIKE HIGH SCHOOL GIRLS!' Which sets the tone of him behaving entirely inappropriately around female students and other teachers. There are a couple of sequences where we see him doing nice things outside school, or the girls meet his kind, if absent-minded, wife, and she tells them what she likes about him, which almost seems like the series is chiding us for judging him on being a creepy pervert. But then it usually deliberately undoes that by having him do something which reaffirms him being creepy.
So I'm not sure what was being driven at there. If it's strictly that the creators think it's funny, and that's it. Or if I'm meant to think that when his wife says she thinks he's cool, but in a way that isn't entirely in fashion these days, that all these weird comments and actions are really him trying to be funny, or hip, and he's just incredibly bad at it. That happens a few times, where he seems to be trying to be funny, and instead is shown to be scaring the hell out of people. But you'd think at some point he'd pick up on the fact he was getting a poor response. Or that the series is poking fun at the part of the audience that also likes high school girls, telling them, "even if you are actually a nice fellow, this is what you're like." But that brings us back around to the sexualizing of Sakaki by the same creators that would be making fun, so ooo. They wouldn't be the first to try eating their cake and having it too, but either way, it ends up making for a lot of scenes that are really awkward.
All that said, there's still quite a bit I enjoy about the series. The friendships we see in particular, both the ones that develop over the course of the series, like Sakaki and Chiyo, and the ones that were already in place that we see the rhythms they have, such as Tomo and Yomi, or Yukari and Nyamo. Even the attempt by Kagura to strike up a friendship with Sakaki, which is always kind of awkward because they share basically no interests, is interesting to me precisely because of how it sputters and stumbles along. The episode where Tomo, Osaka, and Kagura try teaming up to study for finals as The Knuckleheads makes me laugh. The Culture Festival and Sports Fest episodes that come around more than once will reuse certain jokes, but with a different slant, or approached from a different angle, so it ends up working well. The final relay race is one of my favorite bits in the whole series.
For all the jokes or bits that fall flat with me, there are plenty that make me laugh, or at least chuckle. Chiyo getting frustrated and acting more like the child she is, rather than she typically collected prodigy they usually depict her as, always works, precisely because they don't go to that well too often. Chiyo's reaction to Tomo's impression of her, or Chiyo's irritation at being a second-year student getting cooed over by a bunch of first-year students, are the two that leap to mind.