Yes and no.
Personally I will not trust Elyrian banks until they are already tried and true. However if the bank has a backing from the county level or higher, I can see it being potentially a good thing.
Brand new players that come in post launch may have nothing to their name, or may have big dreams and not enough capital to fund it. This is where loans and banks come into play, and I think desperate players will be less picky about their loans when faced with their ideals happening or not.
One a bank begins to get a good reputation and starts to spread it's way into other domains, I can eventually see traditional banking occurring.
Also I do agree that the contract and law system will provide an adequate outlet for banks to be created. If the loaner or loanee breaks the contract then x happens as repercussion. Take the contract to an adjudicator and boom, you can hold the other party liable.
TLDR: I can see some people being desperate enough to not care about trust, and I can see some people willing to take a risk regardless of desperation or trust.