Your understanding of Chevron deference is also incorrect, for what it’s worth.
What in Article I imposes constraints on the presidency? It's almost all about the legislature itself, and hardly mentions the executive. Specifically, I don't see anything that lends credence to the idea that Congress can control the workings of executive departments.
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.
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?
The recent decision to formally overturn this precedent was a reduction in power for the executive branch, since it greatly expanded the scope of when a judge could overrule agencies. However, judges were mostly already doing this, so the big headline ruling was more like a funeral than a murder.
Along with what others have said, you might want to recognize that Congress controls the purse. They dictate how much the executive should spend on each item, though that can be nuanced (as in "up to this much" vs "exactly this much")
That's because the executive does not have the power to create those departments (Article II does not grant that power).
It's an undeniably weird setup: the departments are staffed and run by the executive branch, and they report up into the president, but they are still accountable to Congress, which is where their power to do anything is derived from.