Your sentiments are in the right place, but your econ is way off.
The Federal Reserve "can" function as an arm of the people's government, but that means electing a whole lot smarter class of politicians than we have for the last few decades. Managing an economy is tough work, and we can't expect to get qualified people while paying them about what mid-level retail managers make.
If you glad-handed your way to Congress and were suddenly expected to understand federal finance with only the background your law degree gave you you'd probably be happy to let someone else write legislation involving econ or regulating banks and Wall St. Those "someone else's" always have a dog in the race and will always benefit if they are allowed. This is especially true when it is made easy by a complicit media and financial system to sway voters' opinions against anything the Federal government might do to improve their lives.