Your position is they're not government, but you also talk of them "doing whatever the fed needs to do the job" in response to the fed claiming to be a government agency and within government.

So either you are lying, or the fed is lying, and if the latter your position is that they need to lie to do their job (per your dismissive quip) -- much of which involves transactions of banks drawing on master accounts which are held in part in full faith that they are doing business with a government agency.