This is a great idea (it actually sounds familiar, but not so much that I'll admit to having heard it before). Also, while payday lenders and the like would be the biggest losers, and while [big] banks mostly don't seem to want small depositors, I am most curious to know what the effect would be on banks. There's a lot of inertia in banking, but the USPS has some serious advantages over the big banks even for people not in banking deserts.
People who currently have banks, but only a couple thousand $ in savings, at most, are mostly being screwed over if that money is in a bank. Switching to credit unions broadly makes sense, but not all credit unions are equal, and many people who won't switch offer as reason credit unions' (and small banks') shortage of locations/ATMs. USPS clearly doesn't have that issue.