I disagree with your assessment on the point that congress has control of the fed. Perhaps that is what they say but I think reality is different.
If anything, owing to the broad powers and independence granted to the Fed by the Federal Reserve Act of 1913, the Fed doesn't have to answer to any government agency in any serious way. The Fed Chairman doesn't even have to answer Congress' questions in the oversight committees if it doesn't want to.
The Fed can even refuse to answer to Congress in the oversight committees if it wants to! Any influence Congress has is fairly limited, short of an act of Congress to change the current law.