It's totally a fake problem that untrustworthy people can't be trusted in modern times with bank accounts. The risk is check bouncing. People don't "need" checks anymore though to function in the cashless world.
We already have prepaid card accounts that don't allow you to spend money you don't have. The only catch is that they're run as profiteering machines which extract fees at every turn.
I guess what I'm saying is, let USPS run an operation (like Green Dot, Vanilla, etc.) but with a fee structure based on the true cost of the service only. And the government should give Visa and Mastercard the choice of:
1. antitrust suit to break them up.
2. the USPS cards will be a V/MC, and V/MC will give them very deeply discounted interchange rates
3. a new independent USPS card network with low fees will be built AND commercialized for any issuer to use, and a regulation that anyone who accepts V/MC must also accept that network.
