It seems like you're defining "central banking" in relation to the independence (or maybe ownership of) the central bank. That doesn't seem correct to me.
China has a central bank, but it's fully publicly owned and is not politically independent of the CCP. The ECB, in comparison, is owned by the central banks of constituent banks. And on the other extreme, the Swiss National Bank is publicly traded and a minority of its shares are privately held (i.e., not by governments). (Perhaps related or perhaps not, the Swiss Franc also has historically had very high trust and very low inflation.)
My point is, there are multiple flavors of "central bank" ownership and independence; it seems odd to argue that the primary purpose of central banking is to allow private bankers to violate political oversight when, in some cases, the bank is fully public and not politically independent. Conversely, some examples of significantly more private central banks than the Fed seem to show a history of good management in the public interest.
China has a central bank, but it's fully publicly owned and is not politically independent of the CCP. The ECB, in comparison, is owned by the central banks of constituent banks. And on the other extreme, the Swiss National Bank is publicly traded and a minority of its shares are privately held (i.e., not by governments). (Perhaps related or perhaps not, the Swiss Franc also has historically had very high trust and very low inflation.)
My point is, there are multiple flavors of "central bank" ownership and independence; it seems odd to argue that the primary purpose of central banking is to allow private bankers to violate political oversight when, in some cases, the bank is fully public and not politically independent. Conversely, some examples of significantly more private central banks than the Fed seem to show a history of good management in the public interest.