There was a minor dust-up
after the VP debate
about the constitutional role of the Vice President. Specifically, is the VP a member of the Executive or Legislative branch.
Originally, the VP was person who came in 2nd in the race
for President. This was changed with the passage of the 12th amendment
That implies, to me, that the VP is part of the Executive branch since his/her selection was and remains a function of the election of the President. Since the passage of the 12th amendment, the selection of the Vice-President has been a choice of the Presidential candidate (or perhaps of the political machine of the day). While rare recently, it wasn't uncommon for the President to chose a different running mate for his second (or third and fourth - in the case of FDR) term. (Nelson Rockefeller who got dumped by Gerald Ford, was the last VP to not be invited to run for a second term.)
To say that the VP presides over the Senate and is therefore part of the Legislative branch ignores that all three branches bleed into the other. The President appoints the members of the Supreme Court, the Senate says yea or nay. The Chief Justice presides over an impeachment of a President and Congress tries and convicts. The Supreme Court passes judgement on legislation passed by Congress. This doesn't mean that sometimes the Supreme Court is part of the legislative branch.
There's no contradiction, in my mind, to acknowledge that a member of the Executive Branch presides over the Senate.