No, it doesn't and let's start explaining why.

Besides the first one, there are already "Banks" for everything else you've written. They're called online exchanges. You have to understand that bitcoin was invented to prevent that middleman, not to require him. What would it differentiate with the modern financial system then? You wouldn't need it. For example, you say that it'd be nice if the online bank could help you recover your key. If there are two entities holding the same coins, then there are two owners. The banks can take them whenever they want from you and make their own excuses. 

What stops me from pretending that I got hacked? Even worst, what stops the bank from pretending that I got hacked?