I think it's really significant that Mai and Ty Lee left Azula for Zuko (or at least Mai did, Ty Lee left for Mai) in the context of this fic, there's Zuko, stealing her dolls again. Zuko is the only one of Azula's toys who can claim to be anything like a rival to her. His leaving probably shook her a little, but Zuko not being at court was familiar to her after his exile, and also, it wouldn't have been hard for her to turn it around in her brain into Zuko not being able to hack it at court, the loser. But him showing up and Mai fighting for him? Choosing him over her? Her pathetic brother, who she has always been better than just beat her in something. That as much as Mai and Ty Lee's "betrayal " itself helped shove her off the sanity cliff.
And yes, she was definitely already slipping in "The Southern Raiders." Her affect is much less flat. She grins instead of smirks. She practically starts cackling. She reacts instead of plans. And she goes after Zuko instead of Aang. Eliminating her toy turned rival, and the evidence of her failure is more important than capturing or killing the one person who can stop the Fire Nation conquest. This to me is evidence of her putting her emotional needs over her political best interests. It's the same thing with not killing Zuko for so long in the first place. As long as he was alive, there was the chance that something could happen, and Zuko would end up on the throne, as in fact, he did. If he were safely dead, there would be no other claimant to challenge her. On the other hand, if Zuko is dead, not only does she lose her toy and the last person who loves her, but also, as you said, she loses her fall guy. But he isn't a fall guy for problems external to her as much as he's someone she can feel superior to in her own mind. She has internalized Ozai's view of Zuko as a failure, and so when she needs a self esteem boost, she looks at her brother and thinks "better than that.".
There's a really telling scene in the Southern Raiders where Azula is falling, right before she catches herself with her royal hairpiece (symbolic, given the last tether to sanity, the last prop for her self image she has is her royal father's estimation of her) where Zuko says "she's... not going to make it," and he looks so stunned and so sad. Then she catches herself, and he switches to angry and resentful and says "Of course she did." Azula's right. Zuko loves her, in spite of himself.
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And yes, she was definitely already slipping in "The Southern Raiders." Her affect is much less flat. She grins instead of smirks. She practically starts cackling. She reacts instead of plans. And she goes after Zuko instead of Aang. Eliminating her toy turned rival, and the evidence of her failure is more important than capturing or killing the one person who can stop the Fire Nation conquest. This to me is evidence of her putting her emotional needs over her political best interests. It's the same thing with not killing Zuko for so long in the first place. As long as he was alive, there was the chance that something could happen, and Zuko would end up on the throne, as in fact, he did. If he were safely dead, there would be no other claimant to challenge her. On the other hand, if Zuko is dead, not only does she lose her toy and the last person who loves her, but also, as you said, she loses her fall guy. But he isn't a fall guy for problems external to her as much as he's someone she can feel superior to in her own mind. She has internalized Ozai's view of Zuko as a failure, and so when she needs a self esteem boost, she looks at her brother and thinks "better than that.".
There's a really telling scene in the Southern Raiders where Azula is falling, right before she catches herself with her royal hairpiece (symbolic, given the last tether to sanity, the last prop for her self image she has is her royal father's estimation of her) where Zuko says "she's... not going to make it," and he looks so stunned and so sad. Then she catches herself, and he switches to angry and resentful and says "Of course she did." Azula's right. Zuko loves her, in spite of himself.