I would argue that we shouldn't have to trust the banks. If the banks want to do risky lending, fine - whatever. So far as it's their money (and money they've been given to invest) that they're putting at risk, I have no problem with them spending it all on lottery tickets, using it to build a massive gold statue of their CEO or going to vegas and putting it all on black.
The problem with the banks isn't that they are irresponsible. It's that they're irresponsible with money that isn't theirs and with no comeback when they screw up.
It's never been their money. It's always been our money. Be it savings or debt.
The problem is, we do not care about our money, because of the safety net. In case of bank collapse, someone else has to pay.
In order for that dead lock situation to change, we need to remove the safety net.