I’d question that. This article says the practice had already ended by 1890 when the rules on naming a President pro Tempore were revised.
As I said earlier, if push really came to shove, then the VP could get some concessions. He could note the absence of a quorum at all times and demand that 51 senators be in the chamber. He could note absent members and have the Sargent at Arms compel their attendance or call them to the floor and explain their absence. That would stifle committee meetings which is where all of the work gets done.
He could refuse unanimous consent requests, again requiring 51 senators to overrule his decision.
As someone else mentioned, he could present a non-debatable issue for consideration. Oh, the North Koreans just nuked us and we need to form a response, but the VP says that the President is concerned about the fact that September isn’t National Lawn Mowing Month and that is the issue on the floor right now.
As a Constitutional officer, they couldn’t just vote him out like a Speaker of the House. They would have to realize that he will thwart all business, or at least shape it to his own liking, and negotiate with him to give him some power.
There was a noteworthy incident where Senator Joe “Turncoat” Lieberman was denied an extra minute to speak when Senator Al Franken was presiding. Senator John “Four-flusher” McCain arose to say he’d never seen such a thing before.
But Franken wasn’t using any authority associated with his temporary rostrum seating. He denied a “unanimous consent” request, as any Senator can (though his position made it much simpler to do).
The case of Communist countries (e.g. the Soviet Union, East Germany, North Korea) seem to be an interesting case of legislatures that have broad de jure powers to represent the people (and are often named as such, such as “Assembly of the People”), but in reality they do not engage in meaningful debate or really try to represent the people, they just rubber-stamp whatever the current charismatic leader wants.
The President of the Senate doesn’t have the power to do any of that. A senator, however, can do some of those things, whether they are presiding or not.
The Vice President cannot object, cannot make motions, cannot speak. He can only recognize speakers, rule on points of order, and cast tie-breaking votes. So the type of chaos you suggest would violate the Senate’s rules.
Didn’t Al Gore preside over the Senate during the short time between the 2001 Senate swearing in, around January 2, which was split 50-50 and the inauguration of George W Bush on January 20? The Senate was nominally in Democratic control until Bush took office, replacing Gore with Dick Cheney as Vice President.
But all the Senators would have to do is enact a rule saying the President of the Senate does not have the power to do anything of these things. The Senators can take everything away from him except the title itself and the tie-breaking vote.
The big issue with what Franken did was not necessarily denying unanimous consent. He made the objection as the chair which is a horrible breach of parliamentary protocol and under some rules of order (not the Senate’s however) would not be allowed. What made it even worse and showed Franken’s ignorance of parliamentary law was that he said he was doing it as the junior Senator from Minnesota which means he in effect gave up the Chair without giving up the chair.
What he should have done was recognize that unanimous consent was not given and hope that it didn’t result in a Motion for Division.
Possibly. But then the VP could take the good faith position that the Senate rules then violate the constitution that gives him the power as “president” of the Senate. Every day. And keep those 51 senators here to overrule me. 
If you read the second post in this thread, you may find this was done more than 200 years ago.
Not at all. He’s still the President of the Senate and nobody has taken that away from him.
Now if he tries to claim that the Constitution gives the President of the Senate the power to call for quorums, the Senators will say “Really? You want to show us where it says that?”
Then the next day, the Senators enact a new rule requiring the President of the Senate to be present at the Special Presidential Podium whenever the Senate is in session - and the Special Presidential Podium is located in Baltimore.