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.
It's a blatantly undemocratic power grab, effectively allowing a consortium of private banks to limit Congress's power of the purse. Or at least that was the theory. As we're seeing now, that one putative upside is nonexistent and the Fed is happy to cooperate with Treasury to spend trillions a year. The rentier class appears to be consoling itself with massive asset inflation, while still banking the seigniorage.
The United States could just as easily once again fund all of its spending by creating new U.S. Notes[2] (perhaps without the public debt clause) and then control inflation by extinguishing those liabilities through taxation.
[1] https://www.investopedia.com/terms/s/seigniorage.asp
[2] https://en.wikipedia.org/wiki/United_States_Note