I saw some people talking about how Zuko is extremely ruthless/a horrible friend, etc. and especially as I'm rewatching s3/beginning of s4, I can see that Zuko's childhood deeply traumatized him more than I had realized when watching the show the first (few) times.

First off, the Fire Nation has a strong hierarchy around war/war heros, etc. So even though Zuko was loved and favored by Ursa, he probably felt like he should 'be a man' and that he should be getting that kind of love/respect from his father. The fact that Azula was a lot better at firebending than Zuko was also deeply affected him, as you can see in the beginning of S1, when Zuko is doing a lot of heavy firebending training, ignoring his own health and the health of others. And remember that scene where Azula displayed her firebending to Azulon? Zuko wanted to prove himself so much that he ended up embarrassing himself badly. No wonder it came up in the actual show- it was probably a main event that led to his inferiority complex concerning Azula, and his desperate want to prove himself.

When Zuko was banished and charged with finding the avatar, he probably thought that this was the way he could 'be a man' and finally live up to the love and affection that he though he deserved from Ozai. That was what forced him to become so extreme when it came to finding the avatar, such as putting his crew in danger or being extremely unempathetic. This doesn't just include what we saw in the show, but imagine being on the crew with Zuko for 2-3 years, traveling through the world... and even worse, the South Pole, as they'd never been introduced to an environment like that. With his entire feeling of self-worth reliant on finding the avatar, it was no wonder that he was practically insane over finding the avatar (and I'm pretty sure the crew thought he was insane). Side note, but remember the first time we see Zhao? I mean, Zuko was so defensive! He was so dependent on proving his worth.

You know how Ozai was trying to convince Azulon that Iroh is a disgrace after his loss at Ba Sing Se? And I can see why a lot of the Fire Nation would think so. I mean, you have this great war hero, the Dragon of the West, and after one battle he just becomes some wise tea-loving guy (again, that's probably just what they thought/heard from others)? Most of the Fire Nation not only heard just that from Fire Nation/Ozai propaganda, but a lot of them probably wouldn't be able to comprehend what the loss of his son meant to Iroh. I'm saying this because Zuko probably heard a lot of this growing up, especially from Azula (and because Azula was a better firebender than he was, he probably took a lot of what she said very seriously because of the impact of her other actions) and probably subtle (or less subtle) hints from Ozai or Azulon. When Zuko and Iroh end up traveling together, there's probably some part of Zuko that thinks of Iroh as a disgrace and as being a failure, just like Zuko (leading him to feel even worse about himself, and putting more and more of his self worth onto finding the avatar, as it would also rid him of Iroh, a 'failure'). Iroh's journey and the perception of his journey by Fire Nation propaganda has a big affect on Zuko and on what Zuko believed about himself and Fire Nation societal structure.

Remember in the finale of season 3, when Azula gives Zuko the choice of choosing between Iroh or Azula/the Fire Nation? That stung for everyone-- especially because of the growth that Zuko faced during his time in the Earth Kingdom like releasing Appa and that weird fever that totally changed his mindset. Well, Zuko chose the Fire Nation. For a few minutes, this was a total surprise to me-- as it probably was to everyone else. But that small part of Zuko that though of Iroh as a failure? I can image Zuko thinking "What would I rather do... hide from the fire nation like a-a coward? Like my failure of an Uncle? Or will I return to the Fire Nation and finally get what I wanted from the beginning, my honor?". And again, the part of Zuko that disrespected Iroh was probably sourced from his negative childhood influences, Fire Nation propaganda, and his inferiority complex. So, choosing Azula over Iroh was a decision that a very immature, unwise (and mostly unaware of the deep affects of his childhood trauma) Zuko made.

And finally, not capturing the Avatar. I mean, the amount of times that Zuko said something along the lines of "I'm scared to see Ozai because I didn't capture the avatar!" Even though he helped capture the last major stakeout against the Fire Nation, which is huge! I mean, the fact that he kind of glazed over the fact that he played a major role in conquering Ba Sing Se really goes to show how at that point in time, his honor/the throne didn't even matter to him! All that mattered was capturing the avatar, which really demonstrates the grave psychological impact that the idea of capturing the avatar had on Zuko. I mean, mentally? The fact that he cares more about capturing the avatar than he does about honor shows that maybe he doesn't really care about becoming the Fire Lord or regaining his 'honor'. There must've been a reason for the show to constantly drop little hints of how the Fire Nation ruined families throughout Iroh and Zuko's journey through the Earth Kingdom. Even he doesn't want the throne, Zuko's been so traumatized by the idea of capturing the avatar that his whole self-worth is reliant on it.

*I'm rewatching ATLA right now for the first time in a while, and I'm only up to the the first episode of S4, E1. Once I finish the series I'll probably add this/repost with my more refined ideas-- this is a so-far impressions kind of thing because Zuko's journey has really stood out to me so far.

** I just want to add that I love Iroh and I think that his journey was spectacular. What I say about Zuko thinking of Iroh as a failure is just a theory... because by Fire Nation standards, he was, even if Fire Nation standards are messed up.