There are federal Reserve DC and federal reserve banks.
The Federal Reserve System in DC has thousands of employees, all federal employees. There are seven governors on the Federal Reserve Board of Governors. The Board of Governors is an independent government agency. Chairman, VC and 7 governors are all appointees by US presidents. Federal Reserve System in DC is NOT under the executive branch. The funding is NOT from Department of Treasury. Trump and Elon can't touch them, no buyout or layoff.
On the other hand, the Federal Reserve System has 24 branches across 12 Reserve Banks. The Reserve Banks are the operating arms of the Federal Reserve System. They are private entities. Of course, DOGE can not touch them.