Can Senators Keep Their Senate Seats When Elected President?

Seth Barrett Tillman -- one of the best constitutional law professors not to be a law professor -- argues in this recent paper that McCain, Clinton, and Obama may keep their Senate seats even assuming that one of them becomes President. 

Incompatibility Clause, you cry?  Don't worry, says Tillman: it doesn't apply. 

Here's the abstract:

In a few months, We the People will go to the polls and elect the electors who will elect (or, at least, have an opportunity to elect) the next President of the United States. Short of an act of God or an act of war, it is more likely than not that the next President will be a sitting United States Senator. The expectation is that a Senator/President-elect resigns his or her legislative seat some time prior taking the presidential oath of office. It is widely believed in large and influential academic circles and among the educated public generally that the Constitution requires this result by expressly precluding joint simultaneous Legislative-Executive Branch office holding.

I respectfully dissent. 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.