Its much more complicated than that. There are 12 private regional federal reserve banks in major cities throughout the US, and a government operated federal reserve board of governors in D.C.

But the Federal Open Market Committee (FOMC) actually decides what the future of domestic monetary policy is (i.e. interest rates to private banks, QE, etc.) The funny thing is, private bankers from private institutions are actually on the FOMC. The President of the Federal Reserve Bank of New York (representing private interests, he is appointed by private banks) is the Vice President of the FOMC. There are also 4 other Presidents of Regional Reserve Banks on the FOMC (all from private banks.) And there are 7 members of the Board of Governors (US government representatives) on the FOMC. So thats a total of 5 people elected by the private banks vs 7 people who are appointed by the President and confirmed by the senate. The FOMC decides the actual monetary policy including interest rates to private banks (QE), currency swaps, and asset purchases.

Also, the other Presidents from the Regional Federal Reserve banks can attend FOMC meetings and can contribute ideas, but they cannot vote.

The FOMC is very secretive. They have closed meetings, and many senators would like the meetings at least audio taped because most of the government has no idea what actually goes on in those meetings.

So actually, the Federal Reserve is partially privatized. Basically, the government gets 7 votes, the Private Banks gets 5 votes, and there are 7 other private bankers in attendance at the secret meetings that not even Senators have access too. The Private Banks have nearly an equal say in monetary policy. And guess where the 7 government-appointment Board of Governors get their information from when they are trying to decide whats best for the American economy? The Regional Private Banks. Gee, no conflict of interest there right?

I just got this information straight from the Federal Reserve website: http://www.federalreserve.gov/faqs/about_12593.htm

At least the government isn't trying to hide the truth lol!

Also, Federal Reserve regulators (this is a tangent from the FOMC) overseeing the ethics and business practices of the private banks are notoriously soft and weak, they essentially let the private banks run over the regulators. And the private banks also often hire former Federal Reserve regulators and pay them very well because of their connections to the Federal Reserve (again no conflict of interest right http://dealbook.nytimes.com/2014/11/19/rising-scrutiny-as-banks-hire-from-the-fed/?_r=0)

So yes, while you are technically correct that the POTUS appoints the Federal Reserve Board of Governors, there are many layers to our financial system. Just on the surface, the Private Banks have a nearly equal say (5 banker votes to 7 government votes, plus the other 7 bankers have access to meetings that even the POTUS and senators don't have access too), and Private Banks have found ways to game the system as well by hiring former Federal Reserve regulators and using inside information.