A trusted bank could boost bitcoin adoption for those users who cannot download a bitcoin client and use it. There are a lot of them. Of course, if you are registered here and reading this then you are most likely capable of dealing with bitcoins to a certain extent, but that cannot be told of the general public. Sometimes I don't see why the average user would use bitcoin if they can use (and are used to) banks/cash etc. For some it is just too complicated. An easy application like this may be the solution, maybe not.
A bank is useful for saving/loan of the BTC. It could also be useful for the instant transaction of BTC (same as buying stuff in supermarket with a credit card) which cannot be realized with 10 mins or even shorter block time. As we save the BTC in the bank, the transaction will also not seen on the block chain, thus the bloating of block chain is less a problem.