The legacy banking system has lagged behind in innovation over the last 20-30 years, their time has come and they will be swept aside like the newspaper/media companies of the internet.
“The root problem with conventional currency is all the trust that’s required to make it work. The central bank must be trusted not to debase the currency, but the history of fiat currencies is full of breaches of that trust. Banks must be trusted to hold our money and transfer it electronically, but they lend it out in waves of credit bubbles with barely a fraction in reserve. We have to trust them with our privacy, trust them not to let identity thieves drain our accounts.” [Satoshi Nakamoto. Creator of Bitcoin.]
If you have things like "trusted" miners who must sign the blocks, then you just have a fancy centralized distributed system and there are already plenty databases out there which do this.