A point that's been making the rounds quietly for a very long time is that the Presidency of the United States does not have a great deal of intrinsic power. (Eighth graders that I work with know this.) That's because the job is intrinsically an executive position. His oath of office is that he will faithfully administer the laws. Although the President can propose bills to Congress, his primary legislative authority is negative: the veto. The Founders also made the President the Commander-in-chief of the armed forces, but that too is primarily a negative power used as a check on the generals. The Commander can't even appoint top officers, military or civilian, except by consent of, at least, the Senate. Even within civil administration, the President has few direct responnsibilities; almost everything is assigned by law either to the cabinet secretaries or to independent agencies like FCC, FERC, and so on.
At the same time, of course, the President is also considered the most powerful person on earth. The pertinent question is, How can this be? The answer lies in Teddy Rooseveldt's "Bully Pulpit". The President functions as the leader of the government and, by extension, the leader of the nation. He can at most be a leader; he can't be a dictator. (Some would have liked to, and that hasn't been limited by party affiliation. However, I think those most of the Presidents accused of such desires were really the effective leaders, people who achieved their goals by convincing others to support them.)
What a President can do is foster changes in emphasis. The Constitution requires annual reports to Congress; Presidents have discovered that these reports on the State of the Union can be used as powerful tools for shaping policy. The President also holds the employment of many others in his hands; he appoints and can demand resignations from cabinet officers and commission members throughout the government.
Using his appointment power, executive orders, and public discourse, the President can influence which issues get more attention in the government offices and in the media. In a less polarized environment, the President can shape the national perception of reality. In a more polarized environment, he can at least force selected issues to be kept "on the table" for discussion.
People "on the inside" of business corporations don't really have defensible reputations for knowing reality. Whose clients are walking away? Whose company is selling itself off? And has this ever been seen in before other companies and other industries? Look at any major industry from automobiles to banking; some companies have succeeded, some have failed, and in many cases the failures are clearly tied to obvious factors such as the quality of their products or unsupportable market predictions. My point is that these corporate executives are acting on the basis of belief, which may or may not correspond closely with reality. Just like everybody else.
A common example in the present day debates on economics is the belief that US taxes are driving business decisions. Politicians promote this belief and corporate executives run their companies based on them. A specific case much mentioned now is that the companies may choose to move jobs overseas, "offshoring" predicated on payments required for Social Security and Medicare, unemployment insurance, workers' compensation, and so on. The belief is that these payments overwhelm the advantages of an in-house and on-shore workforce. That may not be true. Sometimes the companies succeed while offshoring, sometimes they run into the ground; this already suggests that tax policy is not a sufficient explanation.
In fact, there is a considerable body of evidence against the idea that the actual tax policy is driving corporate decision making. Belief about tax policy drives decisions. If the President can begin to alter beliefs, then corporate decisions would also change. And the President can influence these beliefs by altering the issues to which corporate decision makers pay the most attention. The President does this through his privileged position in the public debate and by his direct influence over government operations.
The same line of argument can be made about state and local governments, schools, charitable organizations, and so on. The President controls none of them. But he can influence all of them and in doing so the President can shape our belief about what is important enough to pay attention to and thus, indirectly, change actual reality.