The practice of winner-takes-all US Presidential elections developed over time. There is no one date it all started. The plan in the Founders' minds was not fully developed when they started all this. They were developing an entirely new form of government out of pretty thin history. Its phenomenal this government has lasted for so long.
As a practical matter, a winner-takes-all rule works because of the popular picture of the power of government being coherent. The rest of that picture is that a plural rule would be more chaotic. A natural check to the power of government is the fact that a member of the ruling class is subject to election, and must persuade a large number of people they can do the job. Elections lead the populace to an unpleasant sense of uncertainty. A natural result has been a growth in the desire of the people not to want to determine who is the ruler, and a parallel desire to grow the power of government to make decisions for them.
A winner-takes-all-rule in Presidential elections doesn't actually exist. The U.S. Constitution specifies the existence of an Electoral College for determining the Presidential election, but the allegiance of the Electors is up to the several states legislatures according to Article 2, Section 1, Paragraph 2.
Historically, Amendment XII added in 1804 a complication to the process by specifying the President and Vice President were to be selected on different ballots of the College. No where in the US Constitution does it say that these offices are supposed to be filled from a Parties "ticket." But that is the practice. Nonetheless, it would be possible for Electors to fill the Presidency from one party and the Vice Presidency from another, if their state legislature permits.
The states have generally adopted a rule that their portion of the Collegiate Electors are appointed based on the majority vote of that state. Most states have adopted a winner-takes-all system of apportionment. Maine and Nebraska do not use a winner-takes-all system. In 2007, Washington State debated a rule change which would apportion all Electors to the winner of the outcome of the national, not state, plebiscite, but the proposed change was defeated.
Its likely that popular vote is used to determine the affections of the people (and the likelihood of a state legislator's re-election). Only in 1913 was Amendment XVII added specifying the direct election of US Senators.
Only about half of the states have a rule that the Elector must vote according to the popular vote. In other states, the Electors may decide not vote according to the majority rule of the state whose vote they represent. In 2000, Elector Barbara Simmons of Washington D.C. chose not to vote for Albert Gore as she was pledged, as a protest on Washington D.C.'s lack of Congressional vote. An interesting side note is that George Bush did not win the 2000 popular vote but had won in the Electoral College.