I do support the unitary executive theory, but there are limits.
There are things Cabinet/executive branch officials can do that the President could also do under his own authority, such as issuing direct military orders or (in my personal opinion) arresting people for federal crimes.
There are also things Cabinet/executive branch officials can do that the President can’t do. These are actions that draw authority from Congress’s regulatory powers, not the President’s authority to execute the laws. As an example, Congress has delegated to the Federal Power Commission the power to issue rules and regulations affecting or pertaining to the justness and reasonableness of natural gas rates and charges. 15 U.S.C. § 717c(a); upheld in Federal Power Com’n v. Hope Natural Gas Co. , 320 U.S. 592 (1944). In my opinion, the President has no authority to issue those rules and regulations himself; the Federal Power Commission, at least when performing those duties, draw their authority by name from statute and not the President.
So too with the Federal Trade Commission, the Federal Communications Commission, the Food and Drug Administration, etc. You will see that executive “orders” directed at these agencies are often mere suggestions.
~Max