Perhaps a better alternative would be a general credit line available to all Americans managed by the CFPB. The credit limit would be rather small, in the low hundreds, but would be higher for citizens with better credit scores.
Why subsidize loans to this degree? Because there are many, many people that are so poor, payday loans can be their only option, burrowing them into crippling debt, which makes them scarcer consumers (less tax from commerce), economically precarious (more likely to require welfare of some kind), and societally corrosive (what happens when the children of severely indebted parents grow up? Another generation of poverty, most likely).
By all means, let payday lenders do their thing, but they need competiton. Before homeownership policies of the mid 20th century, you would have to be rich to afford a house (when a 50% down payment was normal). Now it's an expectation of middle class life. Competiton is a good thing, and so is letting the government compete.