Well, Yellen recently said something along the lines of 'being part of the team' and Bernanke claimed something similar in his book or talks iirc ( sorry, no references atm ).
The whole independent claim is just smoke and mirrors.
No, its actually legally substantive (though its more independent within the government than independent of the government.) Within its scope of granted authority, the Fed can (and does, regularly) act independently of the President and Congress. OTOH, actions outside of the Fed's legal powers require actions by Congress and/or the Executive Branch.
Now, certainly, the Fed, the President, and the Congress all interact and share their opinions, preferences, etc.
The Dept. of Education's policy decisions are tightly constrained by Congress. If there's a law on the books giving the DoE freedom to determine student loan interest rates, then they can. I don't think such a law exists, though, so the Congress would need to pass one.
The Fed was created by Congress, and has a multi-part mission that it must meet, which has been set by Congress. But beyond that, it has great freedom to determine how to meet that mission on a day-to-day basis.
The Fed is part of the federal government, though. It even has its own federal police force.
However, the Fed often dramatically disagrees with Congress and/or the President when it comes to the best ways of fulfilling their shared values. Witness the consistent complaints from Congress over Fed policies like QE.
> The whole independent claim is just smoke and mirrors.
It's a lot more than that and has a legal basis. If the Fed weren't legally independent, I'm quite sure that Congress would have forced it to adopt different policies than it has.