Well, The Fed is a bit different. It's not exactly a private corporation that is not beholden to the voting public.
More importantly, though, is your note about who this battle is between: it's a battle between groups of people who should both be viewed very skeptically when it comes to the interests of "the rest of us".
Corporate controlled money has been tried in the past; it didn't end well for workers.
Company scrip has a long and nebulous history, its great for tethering your workers to the company town (and company store) while keeping them indebted and unable to leave: https://en.wikipedia.org/wiki/Company_scrip
More importantly, though, is your note about who this battle is between: it's a battle between groups of people who should both be viewed very skeptically when it comes to the interests of "the rest of us".
Corporate controlled money has been tried in the past; it didn't end well for workers.