It will surprise you to hear that we even a big bank which is: A co-op.
There are lots of those types of co-op banks in the US, as well as other large businesses run as the type of co-op you're talking about. That is however a competently different thing than what is being discussed here.
I used to like the idea that the Rabobank is a co-op, but it actually seems to be the least ethical of all the Dutch banks right now. The screwed farmers into taking loans for bad investments, and completely refuse to take their social responsibility seriously, which has become an important thing for banks after 2008.