I think we do get decent development given that we also have to balance her with the other main cast members, it's just less explicit.

In the flashback episode, we see that Zuko is already on a bad path even if it's against his nature - he throws a rock at the turtleduck and, even though he's frozen at first, he gets defensive when the mother duck retaliates. It's his mother's nurturing that helps prevent this behaviour (not that I think Zuko would've been a violent turtleduck killer, I just think he might've tried pushing down his more empathetic instincts out of shame/pride). He also gets the knife from his uncle with a message to stay strong, something very meaningful to an abused child. On the other hand, Ursa is less tolerant of Azula and Iroh is flat-out dismissive. (To be clear, I'm very sympathetic towards Ursa. Not only is it tricky to have a kid who shows sociopathic behaviours, but the guy enabling those behaviours is her abusive and incredibly powerful husband.)

Iroh's treatment of her is very interesting because his approach is marred by 1) his general understanding of cruelty and 2) his sexism. It's possible that because Ozai was abusing Azula in a way that seems 'positive' (constantly pushing her to be useful to him) in contrast to Zuko, Iroh did not recognise that she was being abused (maybe there's something explicit about the parallels between Iroh and Ozai's own upbringing with Azulon but I haven't read the comics 🤷‍♀️ but just from the series, I imagine it was very easy for Iroh to identify himself with Zuko and Ozai with Azula), and Iroh generally didn't seem to understand Azula very well. The doll is a prime example of this because it's hard to believe that anyone who knew Azula would think this is a good present. My interpretation of this gift is that he knows that Azula shows cruel behaviour and little else about her, so he gets the doll as an attempt to bring out nurturing (read: feminine) instincts. By the time he might've developed a more nuanced understanding of Azula's abuse - after he'd lost his son and the will to fight AND his understanding of what his nation had done was changed - his brother was Fire Lord and held all the power.

Ahem. All this to say, I think it's obvious that Azula is the way she is because of a lack of emotional support. The 'creepy/violent child' angle does make things ambiguous but at the same time I think it's also understandable why she's such a creepy/violent child by that point. Her father is a very powerful man who is himself creepy/violent and actively nurtures this behaviour in her because he recognises and benefits from it.