Also if a bitcoin bank is created, who will be the owner of the bank? Who will regulated the bank? Who will audit the bank? A big trust factor is there.
You take things way too seriously. It will be just a service that even you could have started up. Just start up an exchange with a wallet service, where you also happens to give out loans, or allow users to lend Bitcoin to each other. If you so wish you could even offer people interest over the coins that people store in your exchange. Give yourself a fancy name where the word bank is included, and voila, there you have your Bitcoin bank. I have to note that there are already several exchanges that offer a platform where users can lend Bitcoin to each other, so the first steps are already set. And while it's a horrible idea to use exchanges as wallet, it is possible even though it's not convenient. If you look at it from that point, then these exchanges are very close to being a Bitcoin bank.