I agree that it is unlike other corporations, and I'm not really claiming it is privately owned, rather that the Fed and the banks aren't really capitalist institutions. The Fed managers are drawn from the banks and the banks earn dividends from Fed activities, but the appointees are chosen by the state, so they have to appease the regime, too.

I think part of what you're missing about the profit sharing between the banks and treasury is that the Fed's lending goes through the banks, so they inherently make huge profits off of the Cantillion effect, by being the first users of the new money.

I'll also admit "own" was a somewhat poor word choice, because you're obviously correct that they are not allowed to use it however they want. It's a loose way of speaking that can be useful with people who don't realize the banks are direct shareholders in the Fed.

Notice though, that it isn't the Fed banks that are government agencies, it is the Board of Governors. The Fed banks are considered "private" corporations, although neither of us would consider that an accurate label.