Do we really require a bank for bitcoins as we are already well secured with wallet systems. The wallet system is so comfortable and adequate. It reduces our time and energy. We do not require to reach a place or someone for any transactions. All are inbuilt in wallets.
Internet banking is as convenient as the wallets you mention.
If you use a web wallet, then banks can be considered much safer as well.
I am not convinced with the internet banking here in my country, because many banks was breached of hackers easily as their customers are complaining that their account was hacked and from with a maintaining balance it became zero balance as the clients checked it. So I am not really convinced with the security of internet banking and if they will adopt bitcoins, then I will still doubt them.