Your view isn't simplistic enough.
The VP presides over the joint session of Congress because the Constitution explicitly names the VP as the President of the Senate. As far as I know, it is the one and only case where the executive branch of the U.S. government can exert any authority over the legislative branch -- and even then the Constitution only gives the VP a Senate vote to break a tie.
Do not try to mix TOTALLY DIFFERENT functions assigned to the Vice President (VP) by the US Constitution. (USC).
One role as president of only the Senate has nothing crossing over to the much more important role of approving and tabulating electoral votes to select next president.
As I postulated before, there is a reason why the USC specifically assigns the role of tabulating electoral votes to the VP. If it was merely a role to the arithmetic to add votes, that could have been done by any of the Senate or House clerks.