No you don't understand this conversation occupies like half of my brain
Excuse me while I ramble incoherently.
Zuko pretty consistently thinks in terms of 'complete task = result'. (Capture avatar = Dad will love him; get gf food = gf won't be grumpy; light candles for date = date not sad etc.) In this case (following her threat) 'prove I won't hurt the others = Friendship with Katara'. He wants to be friends with all of them, not just allies.
He's done what he was told to, but it hasn't changed anything, what is he missing?
Zuko (and the audience) are under the impression that her anger with him is based in her protectiveness, her caution and her strong sense of morality... BUT THEN THIS HAPPENS
Katara turns around and lets him know EXACTLY what he's missing.
This is not about the others, it's not about him being a threat, it's not about Aang. Its about them! Him and Her!
The rug is pulled out from under our feet. And Zuko's. Katara won't forgive him because she did feel a connection to him, felt like she could trust him, and then he hurt her. Then after she's convinced herself that it was all a deception and doesn't matter, this idiot just shows up and reveals that oh, that wasn't a deception at all, that connection was genuine, but he still betrayed her. The others trusting him grates on her because SHE WAS THERE FIRST AND LOOK WHAT IT GOT HER!
I can't even put it into words how much I think about the line 'Oh, everyone trusts you now? I was the first person to trust you, remember?!'
She's reminding him that their relationship didn't start at the Western Air Temple, it started in the catacombs. Maybe Zuko thinks that who he was then is behind him, but Katara doesn't.
No wonder Zuko's next move is, okay, give me a new, specific task that will fix this problem. There must be something I can do to fix it. He tries by finding Yon Rha, but in the end it isn't his actions that lead to her forgiving him, it's her personal judgement and newfound closure.
I think realising that there's no easy task that can salvage their connection helps him move away from feeling like he can earn relationships from people. When he goes to his uncle (after talking to a certain someone might I add), he doesn't list what he's done right to prove why he should be forgiven (as he does when he joins the gaang), he just apologises and accepts that he might not be forgiven no matter what he does because its not something he can earn through performing services for people.
Altogether, this moment is so special because the show for once is willing to admit that the catacombs moment mattered to these characters and they haven't forgotten that.
Ugh they break my brain they really do.
