Since banks are such important entities, I feel like we cannot leave it in the hands of corperations, so how about non-profit banks?

The intention of a bank (retail/traditional banking) is to act as a intermediary between savers and lenders. Without this intermediary, there are enormous search costs and default risk. The bank is suppose lower search costs and to do due diligence to ensure only loans are made out to credit worthy borrowers. They are providing a service and should be paid accordingly.

One of the "products" these banks could offer is being a sorts of broker between parties to let them get to a lending contract. As compensation, the bank would recieve a certain percentage of the intrest paid.

This is what traditional/retail banking is.

Hopefully that didn't come across as attacking your post, but the 07-08 financial crisis had a lot to do with the non-retail banking sector acting like the retail banking sector without the regulation and capital requirements. Mortgages use to originate from retail banks to the customers, but investment banks began packaging mortgages together and this allowed rates to be reduced (good innovation that helps in home ownership). However, with mortgages being quickly shipped to the next party, (retail) banks became increasing lax (as they would not be holding onto it). Long story short, these products were treated as very safe securities and used as collateral in the overnight market. Lots of incentive issues, regulation, and other problems led to the financial crisis.

The retail banking model is however very standard and I don't think there is any issue with it (although you could make points about being ripped off in fees, etc., I am just referring to the basic framework of retail banking).