“the Federal Reserve is part of the US government”
Actually, the federal reserve banks are semi-private corporations with stock that pays dividends. The government exercises a lot of control over them, but legally they are not part of the government.
The Fed Chairman is nominated by the President, confirmed by Congress, and reports to them periodially. Plus their prophet is turned over to the US Treasury.
Compare that with Mr. Hamilton's bank. Now that was a real "private bank!"