Not trying to sound rude but your interpretation of how Mai calls Azula coward smells like hypocrisy on Mai's end to me. Mai acting all superior over Azula bc she chose to leave what was wearing her down, looking down on her for doing the same, meanwhile Azula was the one who provided her with this opportunity on a silver plate in the first place.
I mean, what's Mai's excuse exactly? She didn't exactly look like whe was about to snap and leave at Omashu, starting a new life. She looked like she knew she was rotting away for months and has been wasting her life for years, yet– what did she do until Azula showed up? What did she say? Did she say a word of objection to her parents, that she didn't want to move to Omashu where she'd be even more miserable? That she's her own person who should be free to decide for herself and she's sick of being their doll daughter/political tool? The difference between Mai and Azula's family fuckeries is that Mai wasn't manipulated and gaslighted into loving her abuser/s, and believing that her value and free will are defined by, where do I even start? Her status in life, her culture, her nation's legacy, her father's, the Fire Lord's will, the list goes on and on and on.
Mai should try walking a mile in Azula's shoes. She should try having Ozai raise her when she has to constantly operate under fear of getting burned, banished, disowned or disclared a wanted outlaw. I'd see how much braver and freer Mai would be in Azula's place, if she had Ozai's mind-fuck no jutsu anchoring her back to her abuser who's like, the only family member she'd have left.
For someone whose character is supposed to revolve around freedom Mai isn't exactly the epitome of freedom herself. Knowing you're being exploited by your family that doesn't appreciate you, only putting on resting bitch face facade as a cover that she's somehow above and not affected by it? Not far from Azula's level of cowardice and patheticness if you ask me.
"Azula avoids her feelings", "Azula never mentions her mother", "Azula doesn't stand up to her father" Boy, sure reminds me of someone else.
But seriously whether they like it or not, Mai and Azula are insanely alike. Mai, unlike Azula, may realize that she's not happy and chooses other people's happiness over her own, but she was just as coward to change that until Azula showed up to pull her out of her misery and provide her with everything she could ask for. How lucky and convenient for her.
To be fair, Mai is not that mature to understand what abuse does to people and the degree Ozai's influence over Azula and why she chooses to do things the way she does. Arguably, canon Mai doesn't really even understand that she herself was mistreated by her parents, so I can actually see her calling Azula a coward either way. Because in Mai's eyes, come on, she's Azula, not a helpless, poor, pitiful meow meow, she knows her better than most people.
Regardless, even if Mai knew and understood the full extent of Azula's "situation" that makes her a coward in her eyes, even if Mai understands why exactly Azula continues to do what she does, she's not going to put up with Azula's bs and be under her heel forever. Mai has self-respect and dignity regardless of Azula's daddy issues or whatever tf she forces people to go along on her missions for.
Mai and Azula are very similar and it's easier to cast stones at your reflection than at yourself