Who would enforce this? The US? Bitcoin is not legally considered a currency anywhere to my knowledge. The bank could in fact just take your bitcoins and not return them. There would be zero legal repercussions.
So we can't have binding contracts in a bitcoin world? That would kinda suck.
More services like My Bitcoin
Sure, the market will decide, but this would be an easy way to steal people's bitcoins.
For a short period of time anyway. Presumably there is more money to be made in offering a service without scamming than with scamming. And whether or not bitcoins are recognized as a currency, there is nothing saying they aren't of any value, so that if you steal them from me and I know who you are, and my philosophy/politics do not go against getting the police involved, that I can't attempt to have you arrested for theft. Presumably a "trusted bitcoin bank" would be transparent as to who they were and seek something like voluntary regulation/certification.
If I have a choice between depositing with a bitcoin bank that tells me they are doing business as such and such in such and such a country and here are the regulations they are operating under versus a bitcoin bank that has none of that, it's not a really hard decision where to park some money.
It seems like you're describing Paypal. It has a worldwide network, an easy to use interface, option of physical card, legal liability, and an online banking system. Apart from fees, there's no reason for anyone who can't back up or remember a password to use Bitcoin at all instead of Paypal.
You're kidding, right? I can think of about 50 reasons why people might want to use BTC rather than paypal that have absolutely nothing to do with passwords, backups, or wallet.dat files.