Right, not "a power of the office".
Yes, it does.
Had I meant that, I would have said "powers", plural. I didn't. I was referring to the power implicit in the office of the President.
If you want a real-world example, all you need to do is look at exactly what happened. President Donald Trump threatened the NFL, and the NFL capitulated. That wasn't because of any specific enumerated power of the presidency. It was because the office itself is powerful. Do you think the commissioner of the NFL would have capitulated so readily to a threat from private citizen Donald Trump? Of course not.
That the American presidency is a powerful office isn’t even remotely a controversial concept.