[1] If the banks, which are financial institutions, can't handle bitcoin, then how come there are exchanges up to now? How come there are exchanges that are like goose laying golden eggs because the owner makes good money from the business? Bitcoin's price might go up and down but a bitcoin bank will just act like exchanges.
[2] I think you meant some "dumb" bitcoin users are saving their bitcoin in an online wallet. Online wallets are only for newbies that don't have nothing yet and should focus on learning the concept of bitcoin first. It's where a newbie would put his early satoshis and then at some point he should move them to a more secure wallet like an offline wallet generated from Electrum.
One thing, you can never trust someone on the Internet. Not even online wallet websites.