We had one [1]. It was considered redundant to private banking. I'm surprised no states have attempted this.
[1] https://en.wikipedia.org/wiki/United_States_Postal_Savings_S...
It's state-owned but functions as a typical retail bank that takes deposits, makes mortgages, business loans, etc.
> Because of our unique structure as a state-owned bank, it is the Bank’s policy not to compete with the private sector for retail deposits. Therefore, convenience products such as debit cards, credit cards or online bill pay are not offered. BND has one location at 1200 Memorial Highway in Bismarck, North Dakota.
This is utterly reasonable. It's a public service, after all.
> and they do not provide online banking
I see an account login button on the top right. You have to open accounts in person. But again, utterly reasonable as an anti-fraud measure.
Which means there is risk in being a bank, as you probably remember from when a few banks failed a little while ago.
But if the fed was a bank, it would be risk free (how could the fed fail?). Which cause all sorts of trouble. The fed could take actions (loan risky money) without worry. Or they would not be permitted to loan money - but that would cause even bigger problems because it would restrict the money supply.
There's no good economic way to do it.