However the theory holds that independent counsels are unconstitutional...
Independent counsel
are unconstitutional. One need not appeal to the unitary executive theory; the separation of powers doctrine is sufficient. The power to investigate and prosecute a crime are completely executive. Therefore, the executive branch alone may exercise this authority. The Congress cannot force the executive to investigate or prosecute, just as the executive cannot force Congress to pass a bill. If the President is refusing to investigate a crime without good reason, then the only remedies are impeachment, and an appeal to the People on election day.
That is merely John Yoo's belief, which, as A18 correctly points out, has nothing to do with the unitary executive doctrine. As its name suggests, the unitary executive doctrine means that the executive branch is unitary--i.e., consists of a single person. Whether the President can or cannot ignore the law is a completely different issue.