See, this is the biggest problem with having these discussions: they require that we all know the facts. The VP does not count the votes. The House and Senate tellers do.
"This implicitly gives him the authority to decide what is a vote and what is not a vote."
So you think that the tellers of the House and Senate have the authority to unilaterally decide who the President of the United States is? That's a scary thought.
If the Vice President has no power in this matter, then it is unnecessary for this job to be performed by the Vice President. It renders the article "of no effect."
This is clearly wrong by established interpretation standards.