Child and successor aren’t mutually exclusive she’s both. Vault hunter’s are expressing empathy, by giving her space to grieve. She’s had all of what, a couple of days to grieve before the story ends and during that time she kinda had the psycho’s who killed Maya and we’re in the process of destroying Pandora to think about. They intentionally left the story on a cliff hanger but I don’t think she’ll get much more than this brief introduction before the next game. I could be wrong though they might tell her story with the second season of dlc.
By your own definition of Mary Sue she still is not one. If Mary Sue’s can’t be said to be less than perfect, then someone who acts like a brat and yells at Lilith for no reason can’t be a Mary Sue. Someone who is a cringy teenager by can’t be a Mary Sue. Ava isn’t a Mary Sue she’s a character who has had a hard life, and that has made her unlikeable to you and others.
You list all the things she’s been “given.” Siren powers, so you’re saying Maya was wrong to choose this girl, why? She hasn’t been given a chance to really show what she can do yet. The keys to sanctuary, that whole scene was set up so Lilith could do her corny passing the torch speech. Lilith is a superhero, they do corny speaches for kids. Children sometimes act like know it alls, look up the Dunning–Kruger effect if you would like to know why they do this.
Letting her lead the charge was actually a brilliant move. She might have been feeling alone in the world when Maya died and she lashed out. Which would be a good reason to run off to your room and be vulnerable in private. Letting her lead the charge says we love you and are here to back you up to a young girl who has only ever trusted Maya. And she did that only after Maya literally offered to take care of her, giving her a place to live, training, food, and support and asking for nothing in return. You may be able to lie with words but a gesture like letting her lead the charge to get revenge for killing her mentor, really says welcome to the family in the most borderlands way.
You want her to show remorse, she has spent her life fighting tooth and nail for everything she needs to survive, forgive her for not admitting she was wrong. It’s not quite so easy to do when you’ve been publicly humiliated for showing any sign of emotion except anger. Which she was at least once in the quest where maliwan guys mock her and read her diary. But probably was a lot more than once judging by the fact she lived in a sewer and had no way to shower. Or don’t forgive her and just continue believing she’s a poorly written character and an entitled brat but she isn’t a Mary Sue.