Larry Sabato has a bizarre article at the BBC. He considers the (unlikely) scenario of a failure of any of the candidates to gain a majority in the Electoral College.
What would really happen if the Electoral College deadlocked in this way is that parties would start challenging close races; if anything has been shown by recent elections it's that the parties will challenge a close race anyway in order to have the ability to beat their opponents over the head with the charge of trying to steal the election. But if this didn't happen, and we worked along constitutional guidelines, there would, contrary to Sabato's claim, be no mess at all.
The election for President would be thrown to the representatives of the people, the House of Representatives; the election for Vice President would be thrown to the Senate, since the VP is the Senate's presiding officer. Sabato tries to make a big deal out of the 'unit rule', i.e., the requirement that the House vote for President not according to representative but according to state delegation. This is hardly a problem. Nobody would be 'disenfranchised' because precisely what we are doing in throwing the election to the House is selecting a President (which is a must) when the votes, already counted, are insufficient to elect anyone President. Everyone has already had their chance to vote; all the votes were already counted; no votes were sufficient to put someone in office; now we need a way to select someone to administer the executive authority of government. This is done by sending the question to the branch of government whose function is to represent the people; namely, the House of Representatives. This ensures that the selection, despite the failure of the general election, still has some connection to the will of the people. But however we may view the President ourselves, the Constitution sets the President up to be Presiding magistrate over the Union of States, so they have the House vote by State. Contrary to Sabato's unsupported claim, there is nothing about the situation that involves "elite control and a lack of popular sovereignty". The whole point is to have a back-up system that minimizes "elite control" and maximizes "popular sovereignty" to the extent possible under an election emergency. There is also nothing in this arrangement itself that will cause a mess; it's about minimizing the mess of an election in which none of the candidates establish legitimacy for the Office of President.
Sabato asks how a President elected in this way can govern effectively; but we can ask this about any excessively tight race. Suppose the President were elected simply by popular vote, and that the race was tight enough that the winner was still within the margin of uncertainty for the election -- i.e., it could not be established definitively that the winner's win was not due to accidents, illegal activities, miscounts, and the like, which in a population of more than 300 million will pile up quite a bit even in a well-run election year. The same question could be asked then; and the same answer given: he or she could govern effectively in the same way any President who came to power under unusual circumstances could govern effectively, namely, by fulfilling his or her constitutional obligations, and proceeding very carefully in matters of policy. And if people don't like the result, there will likely be a constitutional amendment. One might as well freak out about Gerald Ford becoming President despite never having been elected at all. A mess! It was a mess! Well, yes, in the limited sense that it was far from ideal, but we still had a President, and a few years later we had another election. Nobody wants the House to choose the President, but that's why it's the back-up system rather than the main event. The U.S. Constitution is not designed to work only in ideal situations.
You could think up equally 'messy' scenarios. Suppose, for instance, that the President, Vice President, and everyone in the line of succession die all at once. We have no constitutional provision for that! Oh no! It would be a mess! Suppose that the President and Vice President both die, then the Speaker of the House becomes President, and she appoints me as Vice President, then dies. Oh no! A mess! How could I govern effectively?!