Why is the FED the one private organization conservatives believe should be closely regulated?

The FED likes to think they are a private organization but like you just pointed out they are a quasi government agency with officials appointed by the government. If they were private they would pick their own directors and the government would have no input about what they do. Also if you look at their history they really seem to be a straw man for the treasury. One example is their ability to monetize the debt through a third party, scary very scary.