It is true that the Bank of England, the Fed and the ECB have the independent power to set the base rate or short term interest rate. I do not deny that or even propose taking it away. That is not the same as these central banks being independent of the state and government policy.
The Central Banks have Governors chosen and appointed by governments and Parliaments. The Governors have to report to elected assemblies and answer questions and criticisms. They rely on state taxes should they not have enough revenue to pay their costs. They have to work with government or accept government policy on important matters like the issue of government debt, the management of their bond portfolios, and their attitudes towards the actions of commercial banks.
The governments/Parliaments can change the rules, alter the aims, change the budgets and personnel of these Central Banks any time they like. In the UK there have been major changes. Gordon Brown who gave the Bank the power to set base rates at the same time took away their powers to issue government debt and to regulate commercial banks, making the changes a net reduction in their independent actions. The successor Coalition government changed the framework again with major changes to commercial bank regulation. President Biden on taking office appointed several new Board members at the Fed including the two powerful Vice Chairs to change their policies as he was entitled to do, with two leaving the Fed ahead of their retirement date.
The main monetary policy these three Central Banks followed up to 2022 was money printing or Quantitative easing. In the UK every pound of that had to be approved by the government, who underwrote the Bank against any losses. In the last three years all 3 Banks have been pursuing Quantitative tightening. The UK government has been paying huge bills to the Bank to cover their large losses on bonds they have been selling. The ECB has not been selling bonds, presumably because their owners will not pay the losses. The Fed which like the other two overdid the QT had to pump large sums into the markets to prevent a collapse of regional banks. No government can afford to ignore Central Bank actions which help create a fast inflation by creating too much money, or create a recession by tightening too much.
The wrong belief that these Banks are “independent” other than over the base rate means much media and commentary refuses to ask why these 3 allowed or created a large inflation. It means they escape proper scrutiny of what they did wrong and how in the future they could use their power to set the base rate to promote faster growth and lower inflation. They all have to work with their governments.