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I believe the conventional view is mistaken as a matter of the original public meaning of the Constitution. Although the idea of a sitting Senator holding the office of President is somewhat counter-intuitive, this is one example of the dangers of unexamined intuitions. True, the Constitution does preclude joint Legislative Branch-Executive Branch service. But for incompatibility purposes, the President is not part of the Executive Branch; rather, the (elected) President presides over it, as opposed to (appointed) Executive Branch officers -- which are under it. Therefore, a sitting Senator can keep his or her seat while serving as President.
In short, the Incompatibility Clause bars Representatives and Senators from "holding any office under the United States." Here, I (further) argue that the phrase "office under the United States" is a term of art referring to statutory or appointed officers, not to the President. In short, the Incompatibility Clause does not bar joint Senate-Presidential office-holding; it bars Senators from working for the President (or being appointed by the President), it does not bar a Senator from being President.
This article was originally published in University of Pennsylvania Law Review PENNUmbra, which was subsequently renamed University of Pennsylvania Law Review Online.