Charliehall, of course the president is not a king; unlike Louis XIV he is not the state, but he is the executive branch. He does not “work for” or “head” the executive branch, he is it. HHS and Dr Bright work in an executive function, therefore they do NOT work for the people, they work for Donald J Trump, and it is their duty to implement his agenda without question.
That Trump can’t fire individual employees is a matter of civil service protections imposed by Congress. I doubt they are completely constitutional, but so far nobody has tested that. I would guess the best argument for them is that Congress appropriates money for the president to pay his employees on condition that he follows these regulations. But they are his employees, not Congress’s, let alone “the people’s”.
And yes, Bright was removed from his assignment for incompetence and insubordination. I have no reason to believe he is competent, and am certainly not going to take your word for it. Or for anything.