Let's agree that the Federal Reserve is part of the government even while it's an independent agency.
Being independent agencies doesn't make "privately owned corporations" out of the Central Intelligence Agency (CIA, Environmental Protection Agency (EPA), Federal Deposit Insurance Corporation (FDIC), General Services Administration (GSA), National Aeronautics and Space Administration (NASA), Occupational Safety and Health Review Commission, Peace Corps, Securities and Exchange Commission (SEC), Selective Service System, Social Security Administration (SSA), and the United States Postal Service (USPS).
People that work for these independent agencies of the Executive branch are federal employees and their paychecks and terms of service are all administered (just like the military, federal judges, the FBI, etc.) by the Office of Personnel Management --another independent agency. Here's a screenshot from another OPM site:
