Ava is the only real option.

Giving it to Tanis would be awful, tannis is not a people person, she's not a commander. She cant inspire others or whatever.

then, going through everyone else on the ship:

Zero is not a people person, also is an assassin / murderhobo, also is tied to atlas. Not a good choice.

Marcus is an amoral capitalist. the crimson raiders would turn into a for-profit business overnight.

Ellie has shown no real interest in leading and no leadership qualities. She seems to just like being a mechanic

Moxxi is in the same situation as ellie, except bartending instead of mechanics

Hammerlock is likely going home to jakobs, and has always been more about hunting than fighting the good fight

Other vault hunters:

Brick was already kicked out of sanctuary 1 and the crimson raiders on account of him turning a dude's head to paste. Dude's a violent monster and in no way qualified to lead.

Mordecai might have been a good option, but his ties to brick are problematic.

Krieg and Gaige are AWOL and also violent lunatics

Axton is AWOL

Claptrap is incompetent.

I guess there's tina too, but she's basically krieg with explosives

And finally, the player character: So there are two problems with this. Firstly, mechanically, there's no way for lilith to be like "Sanctuary is yours vault hunter. Oh sorry, you picked Zane? I meant Moze. Sanctuary is moze's now. Sorry everyone else" and it would create problems in future story parts if she was like "sanctuary is yours now vault hunter" but then canonically it went to only one of the four, and everyone who wasn't playing that VH would be very confused in the next DLC. There's really no good way to involve the player in this.

Secondly, she doesn't really know much about you other than that you're good at killing and following orders. Like, you stepped off a bus a couple of weeks ago, and just started killing EVERYONE and not much else. You're just as poorly qualified for this as Ava is. But at least if Ava is in charge, that leaves you free to do what lilith KNOWS you're good at: genocide on a planetary scale. If she gave it to you, you'd either be tied up making big descisions instead of out killing bad guys all the time, or it would turn into a one person army, where you make all the descisions, do all the murdering, and everyone else is just along for the ride / under your thumb. That's not what the CR are about, its a family.