From NBC's Ken Strickland
Former parliamentarian Dove is technically correct when he says Vice President Biden, in his capacity as president of the Senate, could overrule the parliamentarian. But the chance of that happening is virtually none.
Dove himself says no vice president has played an active role in the Senate since Humphrey. And here's a historic context as to why as illustrated in Robert Caro bio on LBJ:
After LBJ became VP, when wanted to come back to the Senate and essentially run the Democratic caucus as he had as when he was Senate Majority Leader.
Caro wrote, "[Senator Clint] Anderson said, the Vice President was an official of the Executive Branch. Selection of a member of that branch to preside over a senatorial body would not only shatter the principle of separation of powers but would also make the Senate 'look ridiculous.' "
You might also remember that Senate Republicans flirted with a similar move to end the judicial showdown during Bill Frist's reign as majority leader. But a number of senior Republican senators objected.