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.
Trump does not have the authority to fire or replace any Fed member if they have not violated the Constitution and their term has not ended. He also has no say in choosing who becomes a new Fed member or Fed chairman. He only has the right to nominate candidates but the decision rests with the Senate.
Also, he may have close relationships with members of the Supreme Court, but he does not control them. Likewise, the Fed has 11 members and the rate hike/drop is based on voting results, he will not be able to interfere, let alone control the Fed.
The Fed is an independent institution and its mission is to maintain the strength and dominance of the dollar, not just the US economy.