> It makes sense that we don’t elect a fed chairmen, but if the fed chairmen doesn’t report to any elected official then where do they get their authority?

Congress put in a process for a fed chairman to be appointed in a way that jointly incorporates president, house and senate. Similarly firing them also needs joint cooperation, then the people have some say by electing/unelecting a congress or president that fucks it up. Same for FDA, USPS, etc etc. I definitely want a formalized process to hire and fire these guys and not just up to whims of individual executives.