Well I suppose your question is really - why have a President? It's a good question - particularly when keeping in mind that most countries don't have one. Prime Ministers, like in the UK and Canada, are simply the leaders of their parties in the legislature. So for example, if the United States was to have a system like that, John Boehner would be the head of state.

So then, why did the writers of the constitution choose the Presidential system. It all has to do - not surprisingly - with checks and balances. The Founders - and in particular James Madison- were very concerned with power either becoming concentrated in the hands of one person (like a king), OR power being given to a legislature that just did whatever was popular. The house of representatives, being elected every 2 years, are very susceptible to public pressure. If we were to give them ALL the power, their policy decisions would track very closely to what was simply popular at the time. Why this sounds ok on some level - think what would happen to minority rights. If most people wanted to, for example, ban homosexuality - a legislature would simply institute this (for the fear of not being re-elected). This is why the founders set up both the senate, and the presidency, to be more disconnected from the population - so that they could make harder decisions that weren't necessarily the most popular. Keep in mind that the presidency is actually elected by an "electoral college" that was originally meant to involve upstanding citizens, elected by the people, deciding who ought to be president. Even still - the longer time frame between elections, as well as the ability to stay "above the fray" allows the president to make decisions the legislature may not be willing to make (this is obviously less true in the modern period, but that is a whole other discussion). Similarly, the Senate - up until the early 20th century - was not directly elected, but chosen by state legislatures to represent that state in the Senate for 6 year terms - so again, disconnected from the populace.

So really - the short answer to your question is: the founders felt that a purely legislative system would just do what the majority wanted all the time. They felt the Presidency (as well as the Senate, and in a different sense the Supreme Court) would act in a counter-majoritarian way when needed, protecting fundamental rights and making tough decisions.