Do you think any little part of Azula ever loved Zuko? While manipulating him in season 3 did she ever enjoy his company? Playing volleyball and getting him from the old family house in "The Beach", encouraging him to go to the war meeting sans formal invite and hiding his secret Uncle visits? All those years ago she warned Zuko about Ozai's plan to kill him and tried to get him to spy on Azulon and Ozai. Wishful thinking?
Yes, Azula loved Zuko. You can see her genuinely happy about playing with him as kids, before their mother disappeared and their father started to put more pressure on them both:
I think she still does care about him, in her own way. She’s genuinely concerned about him, here:
Azula: I thought I’d find you here.
Zuko: Those summers we spent here seem so long ago. So much has changed.
Azula: Come down to the beach with me. Come on, this place is depressing.
Azula: So…I hear you’ve been to visit your Uncle Fatso in the
Zuko: That guard told you.
Azula: No, you did. Just now.
Zuko: Okay, you caught me. What is it that you want, Azula?
Azula: Actually, nothing. Believe it or not, I’m looking out for you. If people find out you’ve been to see Uncle, they’ll think you’re plotting with him. Just be careful, dum-dum.
(Listen to the voice acting if you think she’s just being manipulative.)
But that doesn’t mean necessarily mean she won’t try to kill him. When she shot lightning at Katara in the Agni Kai, she accepted the risk that Zuko would take the bullet for her, metaphorically speaking, and didn’t bat an eye. When it comes to maintaining her grip on power, she is absolutely merciless.
One argument I’ve seen flying around the fandom that I disagree with is that Azula’s willingness to use persuasion on Zuko in “The Crossroads of Destiny”, rather than fight against him when she didn’t truly need his help, showed that she cared about him and wanted him home. But here is what many people forget, and the reason why she HAD to get Zuko to come home, regardless of how she personally felt.
Fire Lord Ozai: Iroh is a traitor and your brother Zuko is a failure. I have a task for you…
Does Azula want to capture the Avatar as a big middle finger to Zuko? Of course. But that is NOT her ultimate mission as revealed in the Season 1 finale. She was tasked with hunting down and capturing Iroh and Zuko and bringing them back to the Fire Nation. And unless she returned with both in hand, she would have failed her original mission from her father, even if she’d defeated the Avatar. (And we all know how Ozai deals with failure.)
Now by the end of Season 2, Azula had tried to take Zuko and Iroh prisoner twice before, and each time, they’d escaped because they were a formidable team. They’d had each other’s backs for years in exile, the way we see them in “Winter Solstice, Part 1.” Remember, when Iroh and Zuko fought together in “The Awakening” and again in “The Guru,” they beat Azula plus the highly skilled Royal Guard, and Azula plus the highly skilled Dai Li, all on their own. The reason Zuko got captured in “The Guru” was because he separated from his uncle when the temptation to challenge his sister became too great.
From this, Azula learned that Zuko and Iroh were too much to take on as a united front. However, if she could prey on Zuko’s weakness and get him to abandon Uncle, then Iroh would be a) without combat support and b) less of a flight risk. Sure, it helped to have Zuko as a fall guy in case Aang wasn’t really dead, but that was just icing on the tactical cake. “Divide and conquer” was her surest way of delivering Zuko and Iroh back to her father, just as the Firelord commanded.