Directly electing the president would fundamentally change American elections. The federal government would have to take a much greater role in setting election rules and possibly administering elections. (Elizabeth Warren, during her campaign for the Democratic nomination for president, explicitly linked federalizing elections and abolishing the Electoral College.) Giant metropolises would gain power at the expense of other voters. Splinter parties, spoiler candidates, and billionaires would all enjoy new opportunities in presidential campaigns.
If the purpose of government is simply to put force behind majority will, then the Electoral College is indeed an anachronism. But then so is the U.S. Senate, the federal judiciary, and the Bill of Rights. The late Michael Uhlmann, who first piqued my interest in this question, once wrote an essay entitled “As the Electoral College Goes, So Goes the Constitution.” If the Electoral College is illegitimate, then so are other checks on majoritarianism and limits on the power of the majority’s government.