Okay, here I go.
ATLA was successful for many reasons be it outstanding animation at that time, world-building, etc. But I believe no series will ever be successful if they didn't embrace the naivety of human emotions.
One can't sympathize with say a scientist fails to get funding for his research. A direct confrontation about the character's difficulties without developing an attachment to them would be pretty bizarre. The series never fails at developing this attachment towards the characters, even the antagonists(Zuko, Azula well maybe except Osai).
With that said I think my favorite is the series of moments is what leads to Azula's mental breakdown(I wouldn't like to sound sadistic but you never complained about it earlier so. . .).
Let's start with parent-hood, we could see both Zuko and Azula had a rough childhood(by rough I mean principles taught while you were young and affection towards your children). Love is scarce, Expectations are high but the reward is toxic. Zuko had problems of his own but his mother had his back. When faced with failures she was there to support him. Even straighten his principles if you will(Zuko hurting a turtle-duck). But the same can't be said for Azula love was a reward, not unconditional, toxic. Perfection has its own costs. Becoming obsessed, caring less, leaving emotions behind. She had developed a sort of mental disorder as early as in the "beach" you could tell. Difficulties in social interactions, sadistic tendencies, etc.
On a closer introspection you see that Zuko fared better. I could even say you