I think a central bank would be viable for a monarchy, where the king/emperor is meant to have a good degree of power. Since a monarchy is effectively privatized, it should be reasonably efficient.

Even then, it could be used to screw over the common man, at least it would be for the benefit of the state rather than just bankers.

For a republic, I don't trust the state to keep people's gold safe.

Actually trusting anyone else to keep your gold safe is a pretty bad idea. Keeping your gold in the bank during renaissance Italy could end very badly for you. Song China (gold stored in state depositories) was a bit better I guess. Unless of course, you get a Mongol invasion.