So I think you're misunderstanding what attracts people to Azula, and what makes her stand out among the cast.
Ozai is fundamentally utterly self-interested. Does he actually believe in Fire Imperialism? Sure, but only as an extension of himself and his own power. He is utterly self-interested, and his family, the Fire Nation, and the world itself only matter as an extension of himself and his ego.
Then we have Iroh. Iroh is someone who spent his life doing evil not out of any moral conviction, but because he's someone who goes with the flow and being a genocidal conqueror was the easy path for him go down. After that phase of his life ended, he feels no sense of responsibility for what he has done and no real guilt over his actions, and in fact lacks any real sense of responsibility at all. Sure, he is polite toward others, but that politeness covers up an utter inner vacantness.
Mai is different, in a sense. Mai is someone who believed in the Fire Nation's ideology and had no problem with perpetuating it. She had zero moral or ideological objection to anything the Fire Nation was doing, and indeed she was willingly complacent in most of Azula's actions. What turns Mai against the Fire Nation is not anything having to do with morality, but instead her infatuation with Zuko. Generally, in most societies, people who commit high treason solely out of romantic love for a traitor are not perceived as very positively, but, within in the confines of ATLA, Mai doing so is enough for her to be treated as "100% redeemed."
Ty Lee is a pretty similar case to Mai. She is a little less complacent than Mai is in the Fire Nation's imperialism, but only a little, and, like Mai, she has zero moral objection to anything. Her betrayal of the Fire Nation is for purely personal reasons.
Now we come to Zuko. When you really think about what Zuko does in the series, you realize that morality and ideology are consistently completely irrelevant to Zuko's actions. What he believes in at a moment really just doesn't affect what he chooses to do. Instead, he makes decisions out of personal and emotional reasons. Sometimes that leads to noble actions, but usually it leads to selfish, self-centered ones. In fact, Zuko's decisions tend to be remarkably self-centered and focused on getting what he wants, no matter the cost to others. To the extent he's loyal to anything, he's loyal to people rather than ideas or beliefs, but even that tends to be remarkably limited. He's willing to betray almost anything and anyone to get what he's after, and in fact he spends the series backstabbing one person after another. Fire Nation ideology matters little to Zuko's actions, because he's willing to let the Fire Nation burn if it brings him one step closer to the glorious destiny he so desperately wants. Zuko begins the story believing he has a special destiny that justifies hurting whoever gets in his way and he ends the story believing he has a special destiny that justifies hurting whoever gets in his way; his understanding of his special destiny just shifts.
And then we have Azula, who is utterly different from all these other characters. Azula spends her life in selfless service toward her awful family, her awful Nation, and its awful imperialist ideology. In a real sense, Azula is the only true believer in the Fire Nation's ideology, because she's the only one who actually lets her belief guide her actions and let's it override her self-interest. She is given no reason to reject the belief system she grew up with, so of course she doesn't. Duty, instead of destiny, is her guiding star. This doesn't make her a good person; in fact, it makes her awful, because Fire Imperialism is an awful ideology. However, it does make her very different from the other characters, and potentially very interesting to explore. The other villainous characters in ATLA are villainous due to their vices, while Azula is villainous largely due to her virtues. Young people are supposed to obey their parents, serve their countries, and sacrifice for a greater cause, and that's what Azula does.
That is kind of the crux of the issue. On one hand, we have a lot of Fire Nation characters who get handed "redemption" for behaving selfishly or for utterly rejecting their duty and responsibility in favor for who they have the hots for, and on the other, we have the one Fire Nation character who actually takes duty, responsibility, and service seriously, and she gets "eternally damned" for doing so. That's what stands out about Azula.
Moving beyond that, I agree that Azula's bad actions have nothing to do with mental illness. In fact, I've spent years arguing that.
On the other hand, I think Azula shows every bit as many signs of compassion as Mai, Ty Lee, and Zuko do. She's not empathetic or compassionate to the Fire Nation's victims, but neither are they.
