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.
Checks aren't a thing in most of the world. Opening a bank account is hard because of know-your-customer laws.
The civilised solution to this is very simple: cash! Until you get you bearings and have a bank account opened, just use cash.
Tourist in a country where your debit or credit cards don't reliably work or come with extra fees? Cash!
Buying a second-hand piece of furniture from someone else, and don't feel the need to expose more information than necessary? Cash!
Electronic payment system goes down while you are in the queue at the supermarket? No problem, because you always carry a little cash for backup. (I've had this happen to me several times.)
Cash is the ultimate fallback in civilized society, and worth protecting.
A recent immigrant from countries that don't issue biometric passports (like India) might struggle to open a bank account. I had to devote a lot of time to finding loopholes for my readers.
The status of such migrants is (AIUI) a fraught issue politically, and I'm not sure you can resolve the issues of the unbanked without resolving it.
This sounds like the real problem to solve. It's unacceptable for people to be unable to get a bank account and whatever cards to pay cashless