A national public payment processor in the US would not be more immune to political pressure from religious moralist groups than a duopoly of private processors.

For evidence see, well, all the other institutions of the US federal government.

We have one in Russia, called Mir (world/peace, somewhat ironically). While, yes, there was literally zero downtime on card payments when Visa and MC left the country, they are still irreplaceable for international payments. As a result, everyone who needs to make them or travels a lot has a foreign bank account now, with, you guessed it, a Visa or MC card.

Edit: forgot to say, Mir cards used to work in a few other countries, for example, Turkey and Armenia. But eventually the US government pressured the banks in those countries to stop accepting Mir, because apparently that would somehow help defeat Putin or something.

Then government leaders would be less able to circumvent civil rights and transparency measures (where those exist). That is why the government farms out infrastructure to non government entities.

This way, there is always a threat of businesses deciding not to do business with you by unaccountable forces.

This is what the Digital Euro is supposed to be.