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I haven’t been in the ATLA fandom long enough to know how popular this opinion is, but I think at her core, Azula wants to be accepted, to be loved, to belong.
I think you can tell a lot about a character by finding out what breaks them. Looking at goals can be helpful, but a lot of the times goals can mask true desires.
What truly begins to break Azula is the betrayal from Mai and Ty Lee. She's also strangely bothered by the belief that her mother thinks she's a monster. In the scene where she hallucinates her mother, Azula only breaks down and shatters the mirror when her mother says she loves her. As if Azula desperately wants it to be true but can't believe it, so she lashes out.
In the finale when Ozai leaves her behind to go destroy the earth kingdom, she says: “I thought we were going to do this together” and “you can’t treat me like Zuko”. She’s desperate to belong, to be accepted by her father. And even though she’s had his approval for her entire life, she’s immediately afraid of being cast aside.
While she is pretty obsessed with succeeding at everything she does and doing things perfectly (ie “almost isn’t good enough”), I think the real reason she’s so obsessed is because she believes she must earn her worth in order to be accepted.
Azula hasn't had many experiences with healthy relationships as a child so she makes people stay by instilling fear and proving her worth. Power and success aren't what she truly desires. They are more so a means to an end.
I also think in the last Agni Kai, she breaks down not just because she's defeated, but because Zuko and Katara defeat her together. A painful reminder that other people have support they can rely on, but she has no one.
It's also really interesting to compare her to Zuko because I don't think Zuko has the same motivation at his core. Yes Zuko became obsessed with chasing the Avatar to be accepted by his father, but really it was about his honor. Zuko saw acceptance in the Fire Nation as a means to an end for his own worth and honor.
And that's the reason Zuko isn't satisfied when he's back to belonging in the Fire Nation again in season 3. And he's not satisfied in the Earth Kingdom with his uncle's support, nor satisfied being accepted and trusted by Katara in that cave.
I'm not saying Zuko doesn't care about or want acceptance from people, just that there is something deeper motivating him.
When Azula pushes people away, it's defense mechanism. When Zuko pushes people away, he's yearning for something more.