The Fed is only as independent of government control as regulations permit. The surprising thing about conspiracy theories surrounding it is that the people who entertain and feed them are the same as those who would advocate for less government control of the banking system, not more.
At the time of the Federal Reserve Act (1913) there were over 150 currencies in the US. They ranged from the US dollar (highest security) to state backed and even to individual bank issued currencies. Since those currencies gave the issuers a considerable amount of political power unifying the nation's money around a single issuer was a tough fight in Congress.
Adding to that confusion was the disdain for central bank control of our money supply, even though the fractured system in place made interstate commerce difficult and many unscrupulous players threatened the integrity of the entire dollar based system with land grabs by bankers common in the south and western states.
If you had a more complete understanding of federal finance you would understand that simple dual entry accounting practice and sectoral balances precluded complete control of the money system by the currency issuing federal government. One sector cannot hold both debt and the assets resulting from that debt together.
The federal government needed a clearing bank for debits and receipts within the private sector that had the reassurances of the full faith and credit of the government. That's all the Fed is to our government and any regulatory functions it fulfills are from prior permission by the member banks it oversees.
To retain the permission to issue "credit" (not high power money able to retire that credit that only Congress can create) denominated in US dollars the Fed has to relinquish managerial choice to Congress and the Executive. As an "agency" of the federal government it shares a balance sheet with Treasury across the sectoral divide, so any profits it makes above what expenses it is legally allowed by Congress are balanced against Treasury's debt and made moot.
