The problem here is that I doubt very much Washington would have supported John Adams. It's not just that he & Adams were not particularly close; it was that GW(like most POTUS up to FDR)simply did not
think it was part of his job to tell, or even just advise, Congress on what to do. Besides which helping JA
would not have been easy in any case. JA, a contentious, even abrasive character(Benjamin Franklin, who worked with JA in Europe, once sharply characterized him as "always an honest Man, often a wise one, but sometimes and in some things, absolutely out of his senses")had many enemies in the Senate. One Senator- Ralph Izard of South Carolina- I think spoke for many when- in reference to JA's physical
stature- he bestowed a not-very flattering nickname on him: "His Rotundity"*. It thus seems quite improbable that JA could have led a group who held him in such low regard.
What about his immediate successors? Thomas Jefferson certainly possessed the tact JA so sorely lacked. As leader of the Republicans he was not- unlike many later VP's- a non-entity. But precisely be-
cause of this, most Federalists quite literally hated his guts. They would never have followed him in any-
thing(& when he was VP the Federalists controlled the Senate).Jefferson seems to have realized this; while in the Senate he mostly stayed silent, fleeing for Monticello every chance he got. As for Aaron Burr, being also a prominent Republican he could thus not count on Federalist support. But the Republicans- due of course to Burr's conduct in the 1800 Presidental election- mistrusted Burr & were no friends of his either. Besides, by then the office of VP had degenerated into the total insignificance that would characterize it until roughly the mid 20th century.
The only instance IOTL that I can recall when the VP actually tried to lead the Senate took place in 1975. Nelson Rockefeller, Ford's VP, in order to expedite a vote on a bill to liberalize the Senate's
anti filibuster rule, tried to cut off from speaking Alabama Democrat James B Allen. He justified his move
by declaring that under the Senate rules he had the discretion to ignore inquiries @ the start of a vote.
Nonetheless the backlash from the Senate was so severe Rockefeller was forced back down & apologize
(he was just "a servant of the Senate" he declared.**) If a Rockefeller can't lead the Senate, what hope is there for lesser mortals?
*- Franklin on Adams & "Rotundity" quoted in Alvin M Josephy, Jr., ON THE HILL: A HISTORY OF THE
AMERICAN CONGRESS, pp. 41 & 45 of the 1979, Touchstone edition.
**- Quoted in Jules Witcover, MARATHON: THE PURSUIT OF THE PRESIDENCY 1972-1976, p. 53 of the
1978, Signet paperback edition.