Wouldn't that be like trying to double-spend the same Bitcoin?
This sounds like it just wouldn't work to me.
1) People deposit their BTC into the bank for the same reasons they do today: security/insurance and earnings/interest
2) These people have accepted an IOU for their Bitcoins, the bank can now lend out some but not all of them - just in case someone wants some of them back. This is called their reserve. They can also, in order to stabilize a portion of their resserve, sell certificates of deposit that will cause people to deposit the BTC with them for an agreed to extended time frame, etc. All the same things they do today.
3) The bank lends out let's say 50% of the coins on deposit and keeps 50% in their Trezor ( yes, banks will use Trezor hardware wallets )
4) Assume the people that borrow the Bitcoins spend them all out into the economy
5) Some of the people who get the spent loaned out Bitcoins, let's say 50% of them, deposit the Bitcoins into the same bank
6) The bank now has more Bitcoins on deposit so it can lend out half of them again (!)
7) Rinse, repeat. Every time the loaned out Bitcoins come back as deposits into the bank they get to loan 1/2 of them out again.
Cool, eh? Profitable? Probably. Inevitable? Yes. Inherently evil? I do not think so but some here do.
To a certain extent what is causing our monetary issues is the fact that the banks don't have to wait for the customer deposits in order to make loans. The can borrow money from the central bank, keep some of it and loan out the rest.
Central banking evil? Probably. At least some of the founding fathers of the US thought so.
None of this however has anything to do with having a central bank. What a central bank does is regulate the supply of money. A central bank will make it more difficult to borrow money if too much money is being released into the economy and causing too much inflation. Say for example if it is the central bank's goal to only have $50,400 released into the economy every two weeks, if more money is released into the economy over two weeks then the central bank will make it more difficult to borrow, if less then as much money is released into the economy in as much time then it will make it easier to borrow.