No, I do not agree with the idea of a bitcoin bank. Who will be the authority of the bank?? That is a big question. For a currency which is decentralized and anonymous in nature, bank is not a good idea. While bitcoin is trying to break the conventional rules, people are trying to wrap it within the conventional banking system!
Let bitcoin remain as it is and don't try to make it look like any other conventional currency, because if it happens, bitcoin will loose its charm.
If Bitcoin remained the same as it had been at its inception, we would most likely still see the price in single digits. I understand why people disagree to the idea of a Bitcoin bank, but if we are to be honest, we should agree that the difference between Bitcoin web wallets and Bitcoin banks would be insignificant in respect to the authority that you refer to. So if you are to remain honest, you should be against these wallets as much as against such banks. Besides, if we further follow the policy of truth, we should as well admit that web wallets in fact greatly contributed to Bitcoin popularity and its wider adoption...
But isn't the latter what the most violent and orthodox Bitcoin supporters are looking for?