Sigh. "Executive power" implies the question, execute what, exactly? Whether you take a textualist or originalist view, obviously the answer is, execute the enumerated duties, including "take Care that the Laws be faithfully executed[.]" Equally obviously, "the Laws" are whatever TF Congress says they are, as long as they're within the grant of Article I — especially the Necessary and Proper Clause — and the other explicit grants to Congress in the various amendments.
Oh, and: There's no Necessary and Proper Clause in Article II for the president.
It doesn’t raise any such question because “the executive power” refers to a pre-existing concept. Article II starts: “The executive Power shall be vested in a President of the United States of America.” The use of the definite article “the” without any textual antecedent basis indicates the term refers to a pre-existing concept. In contrast, the use of “a” to refer to the “President” indicates that concept is further defined in the document.
In 1787, “the executive power” was a well understood concept in political theory. The 1653 constitution of the Commonwealth of England had a concept of separation of powers between the legislative and executive. Locke's Second Treatise of Government (1690), and the 1777 English translation of Montesquieu's Spirit of the Laws use the actual term "executive power" dozens of times each.
This is consistent with Article III, which mentions the “judicial power,” but doesn’t mention most of the functions of a court. It doesn’t need to—“the judicial Power” is a reference to a preexisting concept.
The Constitution’s vesting clauses are like the sentence “this operating system runs the virtual memory pager and file system in separate processes.” You know I’m talking about a microkernel without my specifying any further details because the sentence uses well known computer science concepts.
Moreover, whatever “the executive Power” entails, it must always be “vested in a President of the United States.” Congress can alter the extent of “the executive Power” by law, but it can’t take that executive Power—including the power to faithfully execute the law—and give it to someone else.
Then why didn’t the drafters just stop with the first sentence of Article II?