Fun fact: the Chinese central bank is not at all independent, but one-party rule prevents monetary policy from becoming incoherent as a result of short-term political interests tied to the election cycle.

An independent central bank fixes this in a democracy, but now it can exert considerable influence over political outcomes.

Consider that Carter was punished by the public in his re-election campaign for a recession which was ostensibly caused by Federal Reserve policy choices which ultimately benefited Reagan.