Dawn, you seemed to misunderstand my argument when you ascribed to me the position that "we can have constitutional entities not squarely in one of the three constitutional boxes." In the first part of my argument , I tried to make clear that I -- along with the quoted members of the Founding generation -- saw the Vice President to be a legislative official rather than an executive official.
I referred to the Vice President as a "sui generis" office not to suggest that his office is not part of one of the three constitutional branches of federal government but, rather, to distinguish my view from Adam's position that the Vice President is the "Head of the Legislative." I wouldn't say that he's the "Head" of that branch, but I would say that he is "sui generis" among legislative officials: he is nationally-elected, he votes only in certain circumstances, he is first in the line of succession.
By contrast, the Constitution affords him no specified role in the Executive Branch. He has no Executive Power (it's committed to the President); he's not the head of a department. Rather, his only constitutional relationship to the Executive Branch (other than his nominal title) is a prospective one: In certain circumstances, he succeeds to the presidency.
Thus, I think you're hasty to assert that it is "ludicrous" to suggest that the Vice President is Legislative, not Executive, official. John Adams and Oliver Ellsworth -- hardly ignorant of the Constitution -- come out much closer to Cheney's position than to yours.
No doubt, over time the Vice President has come to be known primarily for his role in succession, and for the roles afforded particular Vice Presidents by particular Presidents. Thus, it should come as no surprise that most people would reflexively assert that the Vice President is just an Executive Branch official. I'd recommend, however, that Founding-era discussions strongly suggest that today's conventional wisdom is a little more "conventional" than it is "wise."