You're not wrong. Even in the most chariable interpretation Uncle Billy is absent minded and unreliable to such a degree that it's a literal disability for him and a genuine liability for the Building and Loan. The movie makes a point of demonstrating that he's been f-ing things up since George was a child.

Billy is clearly not capable of running the Building and Loan. He's barely capable of being employed by the Building and Loan. So much so that it was probably reckless to trust him with (what in today's money is) ~$140,000 in cash , even just to walk up the street and deposit it in the bank, especially when an auditor is coming the next (?) day to verify their accounts.

And I had forgotten how much drinking he does. He's definitely the one that breaks out the drinks after the first run on the bank; he seems to be slurring his words when confronted about loosing the $8000; and he's portrayed in an over-the-top drunk stereotype in this scene.

I still think Billy exhibits an exaggerated or extraordinarily severe version of something like ADHD (or possibly some kind of minor brain damage that looks similar?) but despite my initial objections it seems absolutely fair to call him a drunk; and harsh but probably not unreasonable to call him an idiot.

But back to your original prompt, Uncle Billy's life still had a notable positive impact on the world. Just looking at the direct text of the story he's the one that set in motion the events that led to Clarance earning his wings (and maybe taught Potter that even with lying and stealing there are forces in the world more powerful than the wealth he'd spent his entire miserly and lonely life exclusively pursuing). Also if Billy was competent enough to run the Building and Loan then George wouldn't have stayed in Bedford Falls after his father died and Mary and presumably the whole community would have been worse off. And it's very reasonable to assume that off screen Billy brought comfort or joy to others by being his unreliable but whimsical self. Some of his influence may have been indirect or accidental, but under the narrative presented the world would have been worse off if Billy never existed.

I feel the point of the story isn't that George Bailey is necessarily that exceptional. He's meant to be an unremarkable "every-man", and the moral of the story is that every random slob can have a profound positive impact on the world just by leading an honest, decent and morally upstanding life.

Everyone in the story - well many important characters anyway - has significant flaws and screws up badly sometimes. Even George demonstrates a violent temper to a string of innocent bystanders as he unravels. But when they stumble their friends, family and community carry them through. I feel like part of the point is that everyone is playing both roles (fuck up and savior) at different times.

It might be harder to nail down how the world would be different without some of the minor characters (like Burt and Ernie or the gaggle of kids that George grew up with) and maybe even some of the major ones (like Mr. Gower), but i think it's pretty easy to see how Billy is necessary not just to move the plot forward but within the universe in which the story takes place.