Bitcoin is an ideal system that is designed to do without the services of banks. Therefore, I do not understand why complicating this process. Bitcoin Bank will not bring anything for users except for extra costs and control
Have you used a web wallet yourself?
Even if you have, you certainly didn't (try to) use the official Bitcoin client since otherwise you would have known that wallets are not really complicating the process. They are in fact simplifying it by hiding all the gory details from the user's eyes. The user doesn't have to know anything about Bitcoin inner workings, how it is built and of which parts it is made (blockchain, blocksizes, etc). He just need to learn what Bitcoin address is, and that's pretty much all
No doubt the exchanges sometimes face difficulties while dealing with the traditional banks but to my opinion the banks are adopting the bitcoin easily every next day. When we talk about a bank specifically designed for the Bitcoin, it means that we are trying to take things once again to the traditional system. It will be limiting the bitcoin to certain body. With this the bitcoin will lose its increasing number of users as well.