What would change in the Zutara ship/dynamic and/or Zuko and Katara individually if Zuko didn't betray Katara in Ba Sing Se and immediately chose her?
i think most canon divergent zutara fanfictions get it right: they'd probably become close friends in no time, and develop a dynamic similar to what they have in the ember island players. but there's a reason this scenario is best left to fanon - as fun as it would be to see more zutara bonding in the first half of book three, there's always something lost for something gained, and in this case it would likely come at the cost of the depth and intimacy they developed in canon through the WAT and TSR arcs.
it is vitally important to their relationship development that katara gets to be deeply, righteously angry at zuko, and particularly that she goes on her field trip to find yon rha while they're still not on friendly terms. not only does her anger bar her from instinctually falling into a caretaking role with zuko as she does with most of the gaang at one point or another, allowing her to be cared for rather than being the carer, it also frees her from feeling like she needs to fit into any perceived image he might have of her. katara makes it clear to zuko that she owes him nothing - least of all her friendship, and everything that entails.
and it is very this lack of obligation that gives katara the freedom to be wholly and entirely herself. people always point to how katara behaves "uncharacteristically" in the southern raiders to prove that zuko is a bad influence, but the truth is that the way she acts in tsr is an inherent part of who she is. katara can be cold, furious and vengeful just as she can be warm, compassionate and friendly, and the fact that she can freely show both sides to zuko isn't because he's pushing her to be someone she's not, but because she has no need to live up to an idealised version of herself.
this would likely still apply to a degree in a no-betrayal au (tsr would happen in any version of book 3, just because it's so significant to katara's arc), but i find it probable that katara might be more hesitant about bringing zuko along, or less willing to bloodbend before him so readily. katara has to witness zuko's lowest point before she allows him to see hers. she has to take her dark-night-of-the-soul journey with someone she knows has neither the right nor the willingness to condemn her choices, in order to be able to focus entirely on herself and what she needs. very telling that she doesn't ask aang, her future husband, to go with her for support.
it's because zuko allows himself to be a whetstone for the blade of her fury, because he cares enough to find out why, because he tries to help when she's given him no reason to do so, because he stands shoulder-to-shoulder with her at her darkest, most conflicted hour without forcing her to bear the burden of caring what he thinks or feels about it, that katara is able to forgive and befriend him. it is because they see each other at their highest and lowest moments that they're able to have the deepest and most intimate relationship of anyone in the gaang. and none of that would've happened without the betrayal in ba sing se.
after all, love is brightest in the dark.
