#but chiefly the FN royal family's function is antagonistic

[paragonrobits]

from a strictly narrative point of view, the reason Azula doesn't really get fleshed out that much besides a few small elements and characterization details is for the same reason Ozai doesn't either; in terms of what they bring to the story and contributing to the events of Avatar The Last Airbender, they are both designed from the ground up to be antagonists, and not much more than that.

In much the same way that Ozai is a beard-twirling ultra-villain who isn't much more than a spoiled brat and power-mad bully, Azula seems to not have been meant to be much more than that. Narratively, she's clearly built around the idea of being Zuko But Worse, more skilled and deadly than him, more willing to be ruthless and overtly cruel as opposed to Zuko being thoughtless and selfish; there's a reason she is introduced at the same time as Zuko being treated as an enemy of the Fire Nation alongside Iroh, as she supplants Zuko's initial role as a recurring antagonist and most immediate threat.

As a character, she's designed around this purpose; she serves the Fire Nation's interests without question or scruple, her views on the other nations are pretty condescending from what we hear of her (in that the best she can say is if someone acts LIKE a firebender and otherwise she dismisses others as peasants, which mirrors Zuko's insistence that royalty should get whatever they want).

Now, there's certainly room to mirror about the characterization at play. Ozai is personally a pretty emotionally shallow person, which does mirror the real life tyrants and many conquerors; that he's too emotionally sheltered to care about anything but his own wants and regards his children as tools is a fairly straightforward concept. I don't believe it's BAD that he's like this; there are plenty more faceted antagonists in this series, and I don't think there's much benefit in playing the genocidal conquering warmonger as having relatable issues or some kind of point to make. The Fire Nation is EXPLICITLY an antagonist committing unspeakable atrocities in the name of their dominance, self interest and delusions of supremacy. Nuance does not exactly help things besides give ammo to the sorts of people who think the Empire and First Order of Star Wars are the explicit heroes of the story.

An interesting thing in this regard is that the character Azula was originally based on has a pretty minimalistic role in the original draft for the series. There are more large scale changes (Zuko being more motivated by the plight of Fire Nation civilians in poverty from the war, Toph was a boy and part of a love triangle, Iroh was sabotaging Zuko's training) but strikingly Azula's original inspiration is barely present apart from being an antagonist and is barely mentioned elsewhere. Accordingly, its probably likely that Azula's hints of more complex characterization was a very late stage development. (I don't know if 'Katara and Azula are direct foils to one another' was the intended point at that stage but it's certainly the most compelling take.)