In the finale when Ozai told Azula that he would be leaving her behind I think he did it for two reasons: 1) he’s a narcissistic psychopath who just didn’t want to share the glory and 2) after seeing what Azula was capable of (she conquered Ba Sing Se a feat the entire Fire Nation army had been unable to do for the entire war in a matter of days, she came closer to actually killing the Avatar than anyone, and her general bad-assery at firebending) he began to see her as a potential threat to him.
Ozai and Azula were pretty similar. They were both ambitious second born children, that would step over everyone they’d ever met in their drive for power. Their ambition had no bounds. It was implied (and later confirmed) that Ozai plotted to have his own father killed in an attempt to usurp the throne. Azula is a true conqueror and a very dangerous adversary. Though she was fiercely loyal to her father, after witnessing everything she had accomplished and reflecting on his own ambition, who isn’t to say Azula wouldn’t have turned on her father one day for her own aspirations and desire for power.
I think it’s important to note that Ozai did not love his children. He favored Azula because he saw her as a means to an end, but he did not love her. Ozai declared himself the Phoenix King and named Azula the new Fire Lord, a title that became meaningless in the rise of the Phoenix King (all of the Fire Nation flags were immediately covered by flags of the Phoenix King and the palace was left virtually empty). It was a token gift as a “reward” for her loyalty. The significance of the gesture was to demonstrate that any affection Ozai dispensed towards Azula was only as long as she was useful to him.
We can argue all day long about whether Ursa really loved Azula or favored Zuko or if it was intentional but at the end of the day Azula felt like her mother did not love her, and that’s all that matters. Since she felt this she clung onto the remaining parent and nurtured her father’s favor of her. The realization that her father didn’t actually love her and the mere inkling that her mother did (as suggested in her hallucination) would’ve meant that everything she believed for her entire life was a lie. Her inability to accept this was a major factor in her subsequent breakdown.
Hope that answered your question! Kind of really makes me want to go watch season 3 again.
What a coincident that we both posted new stories about Azula receiving injuries same day. Expect that in your story Azula got only her hands burned and while Katara is master waterbender, she couldn't heal her. In my story Azula has full body burns, but Katara who knows little about waterbending will heal Azula completely with only small scars left here and there. Also the time is after war in your fiction while mine is before atla. What a nice contrast!