The dog Hitler's holding in the left picture. Do you think that's a German Shepherd? Or do you think it's a Schaefer, a dog breed I hadn't heard of until I was a tourist in Germany? A schaefer has square hips, not the lowered problematic ones of a shepherd, and is a bit taller than a shepherd. A stronger looking dog.
No doubt the 12 Reserve Banks are private corporations.
It may be of interest to know that they are audited annually by KPMG. Each reserve bank, and the Board of Governors which is quasi-governmental, receive an audit and an audit opinion which is posted on the Board of Governors website. The audit reports are hard to find and they seem to keep moving them within the website, but they're there.
Graph on the right does show the Fed lending a lot of money into circulation. The Fed also bought up a lot of bad debt on its balance sheet. Yellen said she wanted to start selling that stuff back into the market. Don't know if that's still the plan of the new Fed Chairman.
It would be good then to have Bills of Credit, which allowed temporary expansion of the money supply to facilitate individual transactions. Otherwise gold or silver backed currency is a bit too "stiff" and can make trade difficult.
The truth is more interesting.
The money (in the central banking system) is based on debt. Every single dollar out there is borrowed into existence. There is zero (maybe negative) net money supply in the world.
You can create this money yourself. Every time you borrow money, you create money and the corresponding debt. Every time you pay a debt, you extinguish that much money and the corresponding debt.