But my thinking is what about creating a bank which functions the same way as normal banks that trade with fiat currency do but uses only bitcoins as its currency.
Banks dont trade currencies. Exchanges do that. There are plenty exchanges trading alts with bitcoin as currency.
Banks lend money to businesses and rent money for that. and making profit with the difference of that.
For that would be needed lots of only BTC business. that dont exist yet. Most people would rather lend money in currency they expect will stay at least close to what was when he borrow it.
If i am farmer and breed sheeps. i will take loan to buy 100 sheeps, and in 2 years will return 20% more of it. I plan to make 50% profit in that time, so 20% will go to bank and 30% to me.
Would i risk to take loan in Bitcoin. Price might boost 10 times? So i would have to pay 10 times more i loaned, but my profit was just half of what i borrowed.