I feel like he cared about her quite a lot during the show, just not in a positive, older-brother kind of way. I mean, he sees her as a lifelong obstacle that needs to be crushed, humbled, humiliated, put in her place, and stripped of Daddy’s attention, which she supposedly gained unfairly by daring to be more talented and successful at the things their country values.
There are several times when Zuko makes his inferiority complex about Azula pretty obvious, to the point where he fails to even think of her as a human being, let alone as his sibling. Because of that, he doesn't recognize her (also flawed - by no means am I trying to pretend she was a good sister, but at least she made some attempts, and had a few moments of genuine emotional care, some obvious and some subtle) attempts at reconciliation or her desire for some sense of togetherness.
It’s also very notable that he never stops considering his godawful dad family, just like his mom and uncle, but Azula doesn’t even receive that grace. She is treated more like an uncomfortable physical object than a person. But that’s faaaar from ignorance, or from him simply not giving a shit about her on his part.
He mentions her pretty often, both directly and indirectly - when her name comes up, but also when she exists more as a voice in his head representing a certain philosophy, or as a benchmark for bending prowess, skill, and success that he envies. And I think that’s where the irony lies: the core of Zuko’s misunderstandings with Azula is that he never really takes her as her own person. Instead, he projects his own subjective image of her onto her, and then responds to that image rather than to Azula herself - someone who might have feelings, pain, unmet desires, insecurities, and all the other things that he himself, or any living human being, has.
I mean, fair enough, she’s become a master at hiding those things. But in his mind, she is essentially replaced by an image of her that represents something within his own psyche. She represents success, or power, or Daddy's love, or the standards he feels he can never live up to. She always represents something, rather than simply being someone. she doesn’t really get to exist in his mind as a person who lives a real life, with real pleasures and real pain (he even ariculates it in calling her just "lucky", fully internalizing Ozai's abuse of their sibling relationship), or any real obstacles she has to face daily as he himself does (therefore - lack of empathy towards her). She becomes a myth, a mental image that he needs to overcome, rather than a human being and a sister who deserves to be acknowledged and to actually be seen as such by another human being.
Btw, I’m not trying to blame him in this comment. His behavior is explicable, and even logical, given his circumstances. I’m not saying that being understandable makes it excusable (especially by fans, who are supposed to, unlike fire sibs, see the whole picture and didn't grow up under FN's indoctrination), or that it won’t come back to haunt him later, because his supposed solution is ultimately a false one that doesn’t actually resolve the underlying problem. I’m just disagreeing with the original idea that he doesn’t care about her as you can’t really care about someone when you don’t even perceive them as their own person or as a living human being with an inner life. But he does care a lot about the myth of Azula.