Even if they are been chosen by the US president their duty is not to serve their personal interest.
FED 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.
The logic is that we could see someone appointed by the president, and if he doesn't do what the president says, then he can be replaced by someone who will, and if he doesn't do, replace again, until they appoint someone who will.
So there is not really an independence there. The independence is at the power of doing something, FED doesn't require the presidents vote to do something, they are independent in the sense that they can do what they want without asking for permission from him, but they are capable of still getting fired in a sense, not lose their job but replaced. This is why Trump could takeover the whole FED too, just like he did with the supreme court and everything else while he was in power meanwhile.