The US already has a competing payment system for benefits (EBT cards), as do many other countries.
Payments themselves are not a technical challenge, no matter who's doing it. The fundamentals are trivial. You move numbers between accounts.
It's tackling fraud and dealing with disputes that's a challenge.
Isn't US EBT card on the same payment network as credit cards? That doesn't count as an independent system. The same set of "they" as Visa/Mastercard gets the fee.
The hardest part is network effects.
Consider that the largest payment card network on Earth (China UnionPay, 7 billion cards) - decided it was easier just to bootstrap acceptance in the US by a partnership with Discover rather than connecting directly to merchants.
If you want a new scheme to work, distribute something like social security and welfare cheques through it. That immediately forces broad acceptance.