They could let you upload a wallet file and spend from that. Then you could lock away your own copy as your own backup so you don't have to trust them solely.
I don't understand how that would be useful. Once a coin is spent, it cannot be re-spent, right?
While the bank would have access to the funds, they couldn't move them without you being able to know that they did. With eWallet providers today there's no way to audit that they really do have your bitcoins ("all of their deposits") held in reserve:
https://en.bitcoin.it/wiki/EWalletHaving a local copy of that same wallet gives you only a little more control because if at any time you want to spend that wallet, you aren't dependent on that service and they cannot stop you from sending your bitcoins from your wallet under their control to somewhere else.
But you are correct -- having a local copy of your wallet won't help you whatsover if they disappear overnight after transferring the funds from all the customer wallets all at once