The British government owns two notable banks, the Bank of England of course (but for less than a century) and the National Savings & Investment Bank ("NS&I").

The Bank of England doesn't have retail services, once upon a time it offered some products like mortgages to permanent staff but this ceased because it is in a real sense self-dealing, and it's hard to explain to a foreign central bank why their $100M unsecured loan to the bank's Chief Executive, who is coincidentally also brother of the leader of their country is a scandal when the Bank of England is loaning the price of a mid-size home counties house to some mid-ranking employee. I mean, these are different things, but they aren't different enough to make the point, so that programme ended.

NS&I however offers numerous popular retail services for ordinary savers in the UK. It has some conventional savings accounts (obviously paying rather little interest these days), term bonds, some specialist UK products like ISAs, but it also has the "premium bond" which is effectively a not very good savings bond except instead of interest you win "prizes" from the pool of money that would have accrued as interest to all savers. The idea being this is exciting like gambling, but you can't lose (except in the sense that on average you'd make more money with a conventional bond of course).

These are very different activities, they possibly reward different management styles, certainly they would share few resources if they were a single entity.

The idea of using the US Post Office instead is good, because the Post Office already has actual branches all over the country, but it would need to staff up considerably to do this work, so having a separate government entity responsible for the actual bank, with the Post Office handling the front-end experience seems like a sound idea.