There's a reason the Fed is disjointed from the normal branches of government and does not answer to Congress or the President directly. You need that separation to keep the system from corrupting itself. If it were up to Congress, they'd print $100 trillion a year and spend until the system collapsed on itself. Some would argue we've already gotten to that point and we just have yet to see it play out in the economy. That may be true. There's a lot of very real systemic risk in the market right now. Financial mismanagement has destroyed many empires. America's system is vulnerable. Retail banking should be the last of the Fed's concerns at this time. It doesn't cost much, if anything, to open a checking account in the U.S. A lot of the unbanked either have illegal status and are keeping their funds off the books for good reason, or they choose to use cash for privacy reasons, which is totally valid.

If you want a non-profit bank, that's what credit unions are for. They've been around for decades.