A few comments. One, those measures are all Central Bank monetary policy tools, not “the government”. The Central Bank is an independent authority. Two, the Fed has other tools, most importantly trying to influence “expected inflation”, followed by quantitative easing. Finally, and also very important, our elected representatives in Congress also have tools to influence the rate of inflation, by increasing or decreasing public spending and taxation. That has considerable impact on GDP growth and inflation.
In theory true, however the Federal Reserve was created by an act of Congress, can be changed at will by a new congressional act, and its Board of Governors are nominated by the President. Additionally, a large part of the profits of the Fed Reserve go to the Treasury. I think its safe to say its not entirely independent in practice, although probably moreso than most other institutions.
You are of course correct the Fed Res has the tools to combat inflation, but has shown since 2009 that they are reluctant to raise rates in an uncertain economy.