Why do we need an Executive Branch?
Would we be better off with a "Republican Parliamentary" system of government?
Prime Minister instead of President?
Taking it as two separate questions. Second question. Prime Minister is nothing more than an executive just like a president. The British play games with the Prime Minister answering to the Queen. But reality is otherwise. The Prime Minister is the sole leader - the executive.
First question goes back even to how operations were conducted 2000 years before the New World was discovered - before Christ. Any operation conducted without a sole leader in the field was doomed to failure. It was well known that long ago. Same is a problem in Airbus that has multiple leaders and therefore could not avert major manufacturing failures and therefore requires government subsidies. Same mistake was made by the British in 1770s America. Same mistake was made by mental midgets who never learned basic management concepts and imposed their stupidity in Iraq. The fractured American leadership was directly traceable to a supreme commander sitting in the Pentagon where he had no idea of reality and he was divorced from the only thing necessary for success.
Some mistake an executive as a dictator. When the executive is a dictator, then the best one can hope for is stunning short term gains and a long term disaster. Great leaders do not dictate. Great leaders seek out and promote the innovators within their system. See the movie Apollo 13 to understand why great management meant 3 astronauts, who should have died, were saved.
Danger of a sole executive is found in another well understood concept. When they live too long in the ethersphere (typically 10 years), then they become corrupt and disattached. They forget what their job is. Too many yes-men and power grabbers have converged around them. They forget what once made them so successful. They lose touch with what kept them honest.
For this reason, governments also need a Congress, Parliament, Senate, Knesset, Board of Directors, or Dumas. Once a leader becomes a dictator (instead of an advocate of the people), then a committee with power must save that institution from that dictator. How badly did Carly Fiornia harm Hewlett Packard before the board finally removed her? Need we cite Richard Nixon as one of the greatest threats to this nation's government? Other examples were Johnson's wise men who finally got him to realize the disastrous mistake called Vietnam.
A most recent example of why an executive is required is the American management in 2003 Iraq. American soldiers are still dying because the administration violated management principles well proven even 2500 years ago. Worse is that the Congress was even so dumb as to go along with it. Examples include Tom Daschle, Democratic Senate leader. You would think after all these centuries, when stupid management mistakes are obvious, then people in power would see it? But Iraq is the perfect example of management corruption created or advocated by Bremmer, Rumsfeld, Wolfovitz, Feith, George Jr, and especially Cheney.
Management always needs a central leader. But, do not confuse that leader with something corrupt - a dictator. Great management has central leadership that is not a dictatorship. Does the leader work for his people / institution? Or do the people / institution work for the leader? Reasons why are found in what that management must do to advance mankind: innovate.