Ben Kleinerman’s essay is a great example of a structural argument — the kind of argument that I believe, usually, makes the best sense of the Constitution. Here the point is to suggest that independence of the executive would be compromised by a legislature, in effect, supplanting the specially designed electoral college. To the extent that the EC produces uncontested results, I think this is a very strong argument. The 2020 presidential election is an example of one in which the Electoral College results are all certified, and in some cases adjudicated, according to the laws of each state. There is nothing to contest.
But why are the votes counted in the House rather than, for example, by the Supreme Court? One answer might be that it makes more sense for contested Electoral College results to be resolved by the more political branch, since a the winner of a genuine contest or insufficient Electoral vote is a political judgment. This, of course, is more explicitly obvious in the rest of the 12th amendment that provides for the House and Senate to determine the winner of the top two when no candidate for President and Vice President receives a majority of the Electoral College votes.
One of the criticisms of Bush v. Gore is that it did not allow the contested process in Florida to play out in Florida and subsequently, if necessary, in the Congress. It took on a role for itself, in which it certainly did not even hint that the statute that provided for safe harbor dates and so forth was unconstitutional. Thus, it is a safe bet that no Court in the near future would declare that law unconstitutional, and certainly not by a vote of 9-0.