But what exactly are you even asking the banks to do? A green company suffers from the same climate risk as a regular company. Yes, a loan to a green company might reduce the global climate risk, but do you see how you've got a prisoner's dilemma here? A bank that gives a green loan below market rate is basically subsidizing the green transition. That's laudable, but that bank has to compete with other banks that don't make such subsidies.
It almost feels like stating a truism: subsidies should come from governments, not from for-profit corporations.
Maybe you deplore the whole "for-profit" thing, but that's a dangerous direction to go to. It has been tried before, a number of times, and it did not look that rosy in the end.
It's best to work with what you've got: people do business to make a profit. They are not hell bent to destroy the world, but they don't have a motive to save it either. Let them do business, and we as a society we'll take care of saving the world via government action, or via charitable NGOs.