Reply #10: They do make the claim.

[Igel]

They claim that each branch has its own role, and the interaction between them is limited.

The judicial branch should not legislate from the bench; courts are in charge of saying if a law is constitutional (not actually in the Constition, but widely regarded as proper), and saying how the laws properly apply. Congress makes the laws; the executive branch has first-pass interpretation rights and does law enforcement. The court's reasonings are confidential.

The legislative branch legislates, making laws for the population and executive branch. It neither judges nor controls the executive branch directly--there is no provision for 'independent' agencies in the US government, nor room for much input from Congress (apart from being civil). Oversight, like judicial review, is not in the Constitution; it's also widely regarded as proper. Materials used in producing legislation are confidential.

The executive branch should not legislate or judge (military excluded); the executive branch is in charge of executing the laws. And, "The executive Power shall be vested
in a President of the United States of America." Add the implicit "solely" before "vested" and you get the heart of the 'unitary executive' theory as propounded by the developers and advocate of the theory--the idea that the 'executive' is the president is specious (properly it's 'executive' vs 'legislative'), and that the 'unitary' means 'all branches of government under the chief executive' is make believe.

Complicating matters--and making it confusing for those lacking a modicum of good will--is the issue of rights inherent to the presidency. But these are to the side; unitary executive advocates and opponents are split on the matter, with the division being largely partisan (and, presumably, reversible).