The "direct leadership" of the president was important for the very reason the Founding Fathers had intended: A Congress composed of hundreds of members representing different constituencies could not possibly hope to articulate and execute a coherent foreign policy. Under Article II of the Constitution, the President is designated Commander in Chief of the armed forces, and is given the lead role in negotiating treaties with foreign governments. This presidential authority, at least under the terms of the Constitution, is held in balance by the Senate's role in ratifying those treaties and in the Congress's role in determining the budget for the federal government -- money that is needed to fund overseas activities of the U.S. military and Foreign Service.
Considerable debate occurs to this day regarding the balance of powers between the Legislative and Executive branches of government in the conduct of foreign policy, but the Constitution clearly envisions the lead role going to the president. When the American public goes to the polls to elect its presidents, it considers the conduct of the nation's foreign policy in making its decision.
To the extent that forceful leadership on the part of the President of the United States was a major determinant in this country's rise as a global power in the post-World War II world, that was a function of the tremendous industrial capacity of the United States and the strength of its economy at the end of the war, combined with the catastrophic levels of destruction endured by European countries and Japan by the conflict. Even Great Britain, whose level of damage from German bombing raids was serious but which otherwise escaped the ravages of the war, was ill-prepared to resume its role as a major world power. Great Britain's economy was in trouble as a result of the costs associated with the fight against Germany and Japan.
That left the United States and the Soviet Union as the leading world powers as the war drew to a close. American leadership under successive U.S. presidents was decisive in ensuring that, from 1945 to 1989, much of the world was governed by democratic governments and protected from the perceived threat of Soviet expansionism.
The direct leadership of the president in the time that the US was emerging as a superpower in world affairs was “so very important” for much the same reasons that it is today. Largely, it is important because there are no other sources of strong leadership in the US political system.
In the US political system, there is no institutionalized source of leadership other than the president. In many other countries, the political system is parliamentary. What this means is that the head of the government (often called the Prime Minister) is always backed by the political party that he or she leads. It is the party as a whole that is elected to lead the country and the party then picks the Prime Minister. The Prime Minister is not directly elected by the people.
What this means is that the party can be an important source of leadership. It has as much legitimacy as the actual Prime Minister. This is not true in the US. Parties here are not nearly as structured and powerful. Therefore, the president is the only source of forceful leadership available. When the US was rising to power, this mean that the president had to exert personal leadership to steer the country and give it direction.