Article I defines the powers of Congress. Having been assigned to Congress, those powers are not available to the president.
Instead, the president is responsible for faithfully executing the laws. So if the law defines how an agency is staffed and makes policy, the president is bound by that.
If this seems like an imposition upon the president, please remember that the president agrees to these impositions in advance by signing the laws.
> if the law defines how an agency is staffed and makes policy
Strongly contingent upon this, right? Wasn't the whole "administrative state" / "Chevron deference" argument that Congress did the bare minimum in defining what an executive agency is supposed to do, left it up to the executive to direct it as it sees fit? And worse, the supposedly apolitical career civil servants in charge of these agencies may from time to time thwart the will of the democratically elected head of the executive?
Where is the people's recourse then?