The people who say "Let villains be villains.", "Well why not redeem Ozai, he's also a victim of his parents.", or "Azula being redeemed would diminish Zuko's arc." completely miss the point.
"Let villains be villains."
I don't think anyone has a problem with her arc in the main series and her being a villain. Sure, it would have been great if someone (Mai, Ty Lee, Zuko, Iroh) had been in the position to help her, but they weren't, and that's fine by me. Her ending in a semi-ambiguous and tragic way fit well.
The problem is that the authors didn't know what to do with her afterwards and just kept her stagnant without letting her at least develop. I don't want her to "be nice" but in people's attempt to pigeonhole her into being just a "crazy bitch" they forget the positive and sympathetic qualities she does have! It's fine if she stays a villain, or better than hero or villain would be her being an anti-hero, but at least keep her (and everyone else) in line with their series characterisation.
Her world has gone down in flames, her previous relationships are gone (Azula also seems to be scared of changes in relationships, too, so that would be a big thing), and she has to confront her father's wrongdoings and the fact that he never loved her.
So, of course, she's going to change, for better or for worse. Though she can really only go up from there. What else can she do?
"Well, why not redeem Ozai? He's also a victim of his parents, too."
Ozai is also a grown-ass man and a child abuser with zero positive traits that people could see potential in. No, it's not just because Azula is a victim, a girl, or pretty/cool that people sympathise. If Aang had woken up years earlier and the Fire sibling conflict had been between a "Good" Iroh and an "Evil" Ozai as teenagers, then we could talk about a redemption arc, but people seem incapable of grasping that even different villainous characters can have different roles in the story.
"Azula being redeemed would diminish Zuko's arc."
King of redemption arcs my foot. If it can be ruined by his teenage sister also getting one it wasn't that great to begin with. I think it's a much more powerful story to say they were siblings who both had positive qualities but took different paths than to boil it down to Zuko good, Azula bad. Plus, if you must have symbolism and cautionary tales, then Iroh and Ozai or Sozin and Roku serve as perfectly good foils to Zuko and Azula.