The executive departments like Department of State or Education are created by the Congress via legislation. Why can’t the President set up the departments the way he wants like combining education and health and human services or create a new department on his own like the department of outer space exploration and make their principal officer a cabinet member?
A practical matter is that federal funding begins legislatively in the House of Representatives, and a binding budget must be approved by both House and Senate.
Another thing is that permanent aspects of the federal government, unless already created in the Constitution are generally created by legislative acts.
You might enjoy reading this:
http://www.trumanlibrary.org/hoover/ebranch.htm
Truman was able to successfully use ex-president Hoover (widely perceived as an organizational genius) to help him convince Congress to allow him wide latitude to reorganize the various executive branch agencies.
There is something of a precedent for presidents setting up new executive departments in the creation of the EPA by executive order. The order to create the EPA was ratified by congress but didn’t originate there.
Unlike other departments the head of the EPA is not formally a member of the Cabinet, though is considered ‘cabinet-level’.