I think youre getting caught up in the part about "formally addressing the Senate" or that the VP doesn't vote unless there is a tie. The VP if they so choose can sit and determine who has the floor to speak, and rule on decorum. Just because they currently don't (I'm sure they have other more pressing issues to attend to) dosen't mean that they can't. The only time the US President "formally" addresses congress is the Sate of the Union, that doesn't mean that he docent ever speak to Senators.
"It was only after 1953, Vice President Richard M. Nixon changed the vice presidency by moving his chief office from the Capitol to the White House, by directing his attention to executive functions, and by attending Senate sessions only at critical times when his vote, or ruling from the chair, might be necessary. Vice presidents since Nixon's time have followed his example." Before that the office of the VP office was located in the Congressional building, and was staffed and funded via the Legislative Branch
I would also state that the rules of the Senate are phrased "Presiding Officer" the first of which would be President of the Senate, then President Pro Tempore, then whichever jr Senator had been tasked (which is usual).
Nothing states that the VP couldn't go in and do daily functions as the Presiding Officer of the US Senate.
But yes technically she was correct in her statement, but as you see it has become an issue.