Postal money orders already exists, and are exactly as expensive and inconvenient as you would expect.
The actual problem is KYC laws that require payment processors to do the job of the police and refuse service to customers under purposely vague corporate policies without due process, instead of having the police do that job and only have payment processing service denied with a court order and an opportunity to contest it under the law.
What is awful is that I expect it to be inconvenient and awful in the US because there is this idea that anything the government does will inevitably be bad and then there are multiple political parties that try their hardest to make that idea come true when they get elected.
There's no reason for the system to not work in the US other than people trying to sabotage it to the detriment of large parts of the population so their personal tax burden is lower since they don't believe they ever will need the service.
The EU in general doesn't provide government services, the UK does, France does.
When you're already the size of a state, there are basically no more economies of scale to be had by doing something at the size of a union of states. Your fixed costs are already being amortized over something like a million people or more, so the variable costs already dominate and spreading the fixed costs over even more people is past the point of diminishing returns.
But it's not past the point of expanding overhead. So suppose you want to do this in the US -- have the Post Office compete with Mastercard. You propose the bill.
The banks hate it because it's them you're competing with, they have billions of dollars, but more importantly they're a major industry in some states (e.g. New York) which have large numbers of legislators. So they not only use those votes against your bill, they trade their legislators' votes on other bills to get more votes against your bill. Moreover, their legislators are the ones who care most about things like finance, so guess who is on the finance committee.
Then you have states like Virginia where the major industry is defense. They want to destroy privacy protections, because you can't get a lucrative government contract for hoovering up and data mining financial data if you're not allowed to hoover up and data mine financial data. Guess who is on the committees related to that sort of thing. And New York is all for that -- data mining financial data for private purposes is lucrative too, and that amendment also makes it harder to pass the bill they don't like.
Then you have states where the government itself is a major employer, and what they want is more government jobs for their people. So they don't want your program to be efficient, they want it to be labor intensive, which is the opposite. But then it costs more, charges higher fees, requires the users to fill out more forms to use it etc. You may also see private companies looking for the same thing, because they then sell a solution to it, as with Intuit opposing tax simplification.
And so it goes down the line, at each stage making the program worse and harder to pass, until it either dies in committee somewhere or gets passed as some horrifying abomination that makes things worse rather than better.
This isn't some individual proposing to make the bill worse on purpose to demonstrate the ineffectiveness of government, it's a structural issue that produces that outcome systemically.
Now you expect I'm going to suggest having the individual states do it. There are some states that aren't full of big banks or spooks or government bureaucrats, all we need is for one of them to do it. Which would work if they weren't subject to the same problematic federal laws as private enterprise is, created by the same problematic process, which is the reason the market isn't already taking care of this.
In order for the states to do things better, first the feds would have to stop standing on their feet.
Imagine if the post office was getting transaction fees instead of banks.
Since then I've seen post office banking as a way for getting poorer people out from under the thumb of exploitative banks, credit card companies, and check cashing places. Consider as well why should the government give a cut of EBT's cards and unemployment transactions.