as we've recently seen, the president cannot fire the the chairman of the federal reserve. You can nominate them and get confirmed by the Senate, but then  is fully on his own as if running a private corporation.

byzantine

bitcoin is a public money. It's not owned by anyone.

does the federal reserve need to get permission to issue more currency units?

econ

The US Dollar is a private currency

You're wrong, the US dollar is a government currency. The chairman of the Federal Reserve is nominated by the president and confirmed by the senate.

An example of a private currency would be Bitcoin.