That is still a bad idea. For one it violates the separation of powers. You'd have a bank capable of performing open market transactions with the Treasury that is now part of the Executive instead of a quasi-public or private (whatever you want to call it) corporation set up by Congress. Obama couldn't just order the Treasury to issue debt, up to the limits imposed by congress, he could order the NY Federal Reserve Bank to buy that debt at whatever price Obama designated.
"But as you said: the federal reserve banks themselves are private corporations. "
They can be called that because the member banks do own the private corporation. But because of the following differences with normal private corporations, I don't really consider them to be private.
That makes sense too?