Right, the system is a weird hybrid that isn't funded by government. It is funded by profits of their actions like a private entity. I'm not sure what else to call an entity that is funded by its own profits if not a private entity.

From Wikipedia (https://en.wikipedia.org/wiki/Federal_Reserve):

It is governed by the presidentially appointed board of governors or Federal Reserve Board (FRB). Twelve regional Federal Reserve Banks, located in cities throughout the nation, regulate and oversee privately owned commercial banks.

The federal government sets the salaries of the board's seven governors, and it receives all the system's annual profits, after dividends on member banks' capital investments are paid, and an account surplus is maintained. In 2015, the Federal Reserve earned a net income of $100.2 billion and transferred $97.7 billion to the U.S. Treasury.[24] Although an instrument of the US Government, the Federal Reserve System considers itself "an independent central bank because its monetary policy decisions do not have to be approved by the President or anyone else in the executive or legislative branches of government, it does not receive funding appropriated by Congress, and the terms of the members of the board of governors span multiple presidential and congressional terms."

> It is funded by profits of their actions like a private entity.

The USPS is an independent agency of the federal government receiving no taxpayer funding (except for services like postage purchased by other government agencies) that is funded by its own operational revenues.

The parts of the Fed that aren't the Board of Governors aren't really like that, either, though.

> an independent central bank because its monetary policy decisions do not have to be approved by the President or anyone else in the executive or legislative branches of government, it does not receive funding appropriated by Congress, and the terms of the members of the board of governors span multiple presidential and congressional terms.

The first and third are the typical things that define “independent” federal agencies, the second is true of some of them (e.g., USPS).

The structure and composition of the Fed is special though, even in that context, but “private” seems to be misleading behind a false dichotomy.

It is kind of (though this term is charged because of the role corporatism played in the fascist public ideology, though the term is much older and not generally associated with authoritarianism), something of a corporatist institution in and by which state and private power are aligned.