In the context of the hypothetical, Bob and Charlie lend the bank $100k, which the bank keeps in its vault, for whenever the bank examiners stop by to check up on them. They then lend out $1000k as numbers in loan accounts. They put $100k in Alice's loan account, and periodically transfer numbers from Alice's account to the university bursar's account. They do the same thing for Dana, Eddie, Frank, George, Harriet, Iona, Jerry, Karen, and Lisa. Whenever someone comes by to check up on their money, the bank takes them back to the vault and points to the same pile of cash, and they leave satisfied.
Bob and Charlie get maybe 1% interest on their $100k, while the bank charges 10% on all $1000k of the leveraged loans. At year's end, the bank has paid B and C about $1k, and collected from A D E F G H I J K and L about $100k. If Bob, Charlie, or the bursar ever unexpectedly come around to make a withdrawal, the bank phones up the national central bank and says, "Yo. Boss says you need to come by and do that thing we talked about that one time. It's important." Then a truck shows up with very important pieces of fancy paper that the bank can give to people, so that they go away without getting mad. It's very important that no bank ever runs out of those, because then people wouldn't give them back to the banks to be reused, by giving them to other depositors!
It's all very complicated and important and not crooked at all. You can certainly trust those banks, because all their employees are clean and smell nice and speak clearly and wear good-looking suits.
But it is also important to realize that the bank does not need to know Bob or Charlie very well. It isn't really even strictly necessary that Bob or Charlie be real people. All that is necessary is that they have entrusted a symbol of their savings to the bank, to be used as the bank sees fit, without any direct oversight, in exchange for a pittance in interest. This is really only a good deal if you have so much money that your investments would otherwise overwhelm the capability of businesses to generate a good return on it. So banks have a few customers that deposit enormous quantities of cash with them, because the only other entities that can actually pay out a 3% return on that much dosh are national governments, state-partnered enterprises, and multinational mega-corporations. Bob and Charlie represent round-off errors. But there are a lot of Bobs and Charlies, and in aggregate they do help the bank make money, so they are tolerated, with a token amount of ass-kissing to keep them happy. And if one gets screwed over by accident once in a while, that's no big deal (to the bank).
The bank does need to know Alice and D E F G H I J K and L, but only just enough to figure out how much extra they need to charge to cover their default risk. The bank's major concern is that one of those people might actually be Maurice, who plans to default, but he is untouchable, because the bank does not know who he really is. But they know who he was pretending to be, so screw that guy; he shouldn't have let someone else pretend to be him without his knowledge, right?
So I'm not really certain that knowing all the insignificant peons would actually help. The fundamental problem with banking is that the (remaining) individual firms are too large for the median customer to matter to them at all. They have no intrinsic economic incentive to even acknowledge that a problem exists, much less admit responsibility for it, because their real business is aggregating investment opportunities that are too small to bother with individually into packages large enough to interest investors with colossal amounts of cash, and taking a percentage off the top.