The vice presidency
What’s wrong with this argument is that there’s nothing unique about the Vice-Presidency in this respect. The President, for example, has the right to veto legislation; why doesn’t that count as his likewise exercising a legislative function? The President also appoints the members of the Supreme Court; does this mean he exercises judicial functions too? Of course the Senate can nix the President’s judicial appointments (thus likewise exercising judicial functions), as well as nixing his Cabinet appointments (thus taking over executive functions). Congress can also impeach the President (thereby intruding into both the executive and judicial spheres). The Supreme Court for its part can strike down unconstitutional legislation (thus exercising a legislative function). And so on.
If the Vice-President is not part of the executive branch, then by the same logic the President is not part of the executive, Congress is not part of the legislative, and the Supreme Court is not part of the judicial. Which seems rather a reductio ad absurdum.
This letter has not been edited.