Right, and the executive power is the power to execute the laws that Congress writes (plus foreign policy, armed forces, and a bunch of procedural stuff — Constitutionally quite weak actually [by design])
Well, yes, the office of president was created to be weak relative to the British monarchy. But the substance of executive power (i.e., what actions are authorized) isn't really the issue, but rather whether anyone other than the president has the constitutional authority to do those things.
Take, for instance, the executive power "to grant Reprieves and Pardons for Offences against the United States" (Art. 2, Sec. 2). There is a pardon attorney who advises the president, but it is solely the president who has the executive power to grant the pardon; in that sense the president exercises the pardon power exclusively (or phrased differently "to the absolute exclusion of others").
Yes, Congress writes the laws, and the executive executes them. That’s certainly true in the general case. But when we talk about independent agencies like the FCC, the relevant question isn’t just the functional division of labor, but rather who holds the constitutional authority versus who exercises it under statutory constraints. In other words, even if Congress intends for the agency to act independently, the president’s Article II authority still provides the baseline for executive power. The pardon example illustrates the principle that some executive functions are exclusively presidential; independent agencies are essentially a statutory modification of that baseline, not a negation of it.
Of if one prefers to have it from directly from Chief Justice John Marshall, Marbury v. Madison, 5 U.S. 137 (1803):
"By the Constitution of the United States, the President is invested with certain important political powers, in the exercise of which he is to use his own discretion, and is accountable only to his country in his political character and to his own conscience. To aid him in the performance of these duties, he is authorized to appoint certain officers, who act by his authority and in conformity with his orders.
In such cases, their acts are his acts; and whatever opinion may be entertained of the manner in which executive discretion may be used, still there exists, and can exist, no power to control that discretion. The subjects are political. They respect the nation, not individual rights, and, being entrusted to the Executive, the decision of the Executive is conclusive. The application of this remark will be perceived by adverting to the act of Congress for establishing the Department of Foreign Affairs. This officer, as his duties were prescribed by that act, is to conform precisely to the will of the President. He is the mere organ by whom that will is communicated."
That is the power that’s vested in the executive