That's the popular mythology. It's purpose was to elect an endless succession of George Washingtons.
This is exactly right. Anyone who puts forth the argument the purpose of the Electoral College is to somehow balance large vs. small states, or urban vs. rural influence hasn’t read their Federalist.
The EC was established in order to put a - somewhat permeable - barrier between the will of the people, and the person of the President. The Framers felt the people should be listened to, but they could not be trusted to make the final judgement.
In Hamilton’s own words:
“It was desirable that the sense of the people should operate in the choice of the person to whom so important a trust was to be confided. ...
It was equally desirable, that the immediate election should be made by men most capable of analyzing the qualities adapted to the station, and acting under circumstances favorable to deliberation, and to a judicious combination of all the reasons and inducements which were proper to govern their choice. A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations.”
The Electoral College that currently exists - one where Electors are more-or-less completely bound to choose whoever won the popular vote in their State - would horrify Hamilton, Jay, Jefferson, and the rest of the Framers.
Hamilton goes on to speak of the benefit of “this detached and divided situation” which would ensure “the office of President will never fall to the lot of any man who is not in an eminent degree endowed with the requisite qualifications;” and men with “talents for low intrigue, and the little arts of popularity,” would never become President.
I wonder how Hamilton would feel about some of the men who have filled the Oval Office in the 20th and 21st Centuries.