Independent FED is definitely important, but independent doesn't mean without any agenda. Sure, no president has any power over FED, that is an awesome thing, and every FED in the whole world should be independent like that.
FED is not exactly independent though. The chair of FED is nominated by the US president and is approved by the Senate. Not to mention that the US President has options to remove the chair even if he can't directly fire him. I don't think we can call it independence.
Even if they are been chosen by the US president their duty is not to serve their personal interest.
FED chairman's duty is to serve the interest of economy in their country and President is out of scope of any administration.
But somehow we cannot deny that their our corrupt officials out there have debt of gratitude towards their appointments gotten from the President. So hopefully the chosen official have right or solid principles and cannot be controlled by anyone.