Finally, we get to this question of whether central banks are in fact independent at all. The 80s and 90s were the high point of central bank independence globally, but since the that time and particularly since the great financial crisis, central banks implicit mandate to maintain economic stability has resulted in more direct political control over its function.
Things like bailouts, targeted monetary policy, the different distributional impacts of monetary policy, the pressure on central banks to be more transparent about their goals and objectives. These are all political dimensions of central banking, and in fact maintaining the stability of the financial system as a whole, is often at odds with the imperative to maintain price stability or control inflation, which is another mandate of central banks.
How that trade-off is made is a political decision.