This sounds like a problem with immediate impact which will eventually (1-2 years?) go away. Creating a new payment system, deploying it widely, getting people to use it, all takes way longer than that time period.
They could build on the Direct Express debit Mastercard system that already exists to give social security (and other federal payers) a place to direct deposit funds for people who don't have a bank account.
You'd need a way to get money on the card, but that could be ATM deposits, presumably.