in the Presence of the Senate and House of Representatives

This clause is what gives Congress the ability to vote and resolve any disputes.

If I look hard enough I can see your interpretation. However, "presence" is such a weak word that if the framers had intended any sort of authority, they would have used a stronger word.

The very next clause says

The Congress may determine the Time of chusing the Electors

"May determine" is much stronger than "in the presence of".

Perhaps "in the Presence of the Senate and House of Representatives, which may resolve any disputes"? Except that the president of the Senate already has authority over the counting. So I don't think the witnesses have any authority.