So, theoretical question: If I was a wise king in ye olden days when precious metal coins were still the money, would it make sense to establish a sort of “central bank” with the mandate to manage money?

To make sure that each coin has exactly X grams of gold/silver/copper/etc in it, according to its stated value. You know, since kings had a habit of cheating on that, and markets dont appreciate those sorts of shenanigans.

And probably also to hunt down money-based crimes.