This should bring out a variety of answers. I would say that there is a both a practical and symbolic rationale behind why the leadership of the President is so very important. Practically speaking, the President is a symbol for their political party. This carries with it a great deal of weight and recognition for this party is going to be subject to name recognition with the President. If the President is successful, the party will stand to benefit from that success, as well. Conversely, if the President is unsuccessful, the party suffers. This can end up translating to altering the balance in the two party system of the Legislative Branch in terms of number of seats for each party in midterm elections. The President's rapport with Congressional leaders can also be vital for the passage and introduction of legislation, which again is reflective of the President's standing. If the direct leadership of the President is received in a very positive way, the legislation they propose and for which there is advocacy stands a greater chance of passage. The President's political capital becomes vitally important in a practical setting here. If the President enjoys little political capital, their direct leadership is hampered significantly. From a symbolic standpoint, the President of the United States is tabbed with the cliche, "leader of the free world." In this instance, there is some truth to the cliche. The President is an international symbol. Whatever else may come of the nation and its prestige amongst the world countries, there is a certain political value when a nation "hosts" the President of the United States or when there is a sit down between a leader of a country and the President of the United States. This perception is also felt on a domestic level. The perception of the President on the part of their fellow American citizens can be as important, sometimes more important, than what is actually done. If the President is perceived to be weak and ineffective, then there is a good chance that their direct leadership will suffer. The President's perception plays a critical role in their ability to govern. This is a part of the component of direct leadership, and an essential leader why it is so important in the United States.
When I teach US Government, I always ask my students to look at the Constitution to see what powers the President actually has. We expect him (so far) to be very strong and important as a leader even though he has very few powers. So why is this?
To me, this is because the President is the only official who is able to command national attention from pretty much everyone (not just those who follow politics closely). No other official can get himself on TV whenever he wants. No other official can make headlines with just about any speech he gives. So the President is really important because he is the one who can do the most to set the agenda. He can set out a bunch of policy goals and people will pay attention.
He is also the one with what Theodore Roosevelt called the "bully pulpit." He can use it to set the agenda, but he can also use it to try to persuade the people to support his agenda. No other official has this power.