It's become plain to me after all this time that America has too often been saddled with teams of duds in the executive branch. A President too often gets elected surrounded by yes-men and "handlers," and the Vice President has nothing to do except suck up, travel around and get photo ops (don't tell me he really does actual diplomacy or something, it's not even in his Constitutional job description.)
The Twelfth Amendment modified the procedure for electing Presidents in 1804 as follows:
"The electors shall meet in their respective states and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and of all persons voted for as Vice-President...."
See http://www.law.cornell.edu/constitution/constitution.amendmentxii.html for the rest of it, but what that means is the President and Vice President since 1804 have been elected separately, each candidate campaigning for either one or the other office but not both. As a result the political parties developed what's called the dual ticket system, in which the slate put forth for the executive branch is two persons -- one for President, the other for Vice President, and the primary system these days (the caucus system before that) picks the Presidential candidate, and then they negotiate over who gets the Vice Presidential slot, with what I consider questionable results, especially recently.
After the Twelfth Amendment passed, I'm sure historians note that the quality of leadership in the country, with certain glorious exceptions, underwent a gradual decline since the first decade of the 19th century and has never really recovered. Admittedly it is hard to find national leaders of the stature of Washington and Jefferson in any era or nation, but we managed an era of excellence in leadership in the nascent decades of the United States. Not so much afterwards. So what was it we were doing right, before 1804? And why did it work so well then compared to later?
The Constitutional framers originally set up the system for electing the President and Vice President as follows:
"The electors shall meet in their respective states, and vote by ballot for two persons, of whom one at least shall not be an inhabitant of the same state with themselves... The person having the greatest number of votes shall be the President, if such number be a majority of the whole number of electors appointed... after the choice of the President, the person having the greatest number of votes of the electors shall be the Vice President...."
That means the electors chose two persons for the ballot for both offices together, the winner became the President, and the runner up became the Vice President. That meant that in George Washington's case, John Adams, the runner up in votes, became Vice President, and when John Adams was elected President, the runner up was Thomas Jefferson. In each case the two most capable men (as deemed by the Electoral College as a whole) were always selected for those two offices. If the runner up just happened to be in the loyal opposition, too bad, which is why the Democratic-Republicans changed it in 1804 with the Twelfth Amendment. But, can anyone argue that the presence of the opposition in the person of Thomas Jefferson kept John Adams on his toes when Adams signed bills like the Alien and Sedition Act into law? Contrast that situation with the next Vice President who happened to share the same party with his President, Jefferson, who entered the office already seasoned with experience in the executive branch. Burr was certainly popular once, but with the one-party system in full power, he was no Alexander Hamilton. The facts are that the Constitution was changed to a two-ticket system under a political party that had a supermajority in Congress and wanted nothing in its way, including potential trouble from Federalists if they ever got popular enough to win second place again, which they did within months of the Amendment becoming law.
What I believe this change destroyed was a potential and crucial role for the Vice President as an ombudsman and a critic of the Administration. Imagine the value of a Vice President Hubert Humphrey weighing in on some of Nixon's more controversial policies in Vietnam or regarding domestic surveillance or the principle of executive privilege. Would Kennedy have been assassinated if Nixon, Kennedy's best Republican friend, were waiting in the wings instead of his bitter internecine rival Johnson? Or how about Wendell Wilkie or Hoover in the 1930s, possibly keeping Roosevelt from veering far to the left and endangering the economic recovery in the Depression, or Jimmy Carter pulled centerward by Ford, or Reagan pulled centerward from the other direction by Carter? Can it be denied that George H.W. Bush would have better been served by a Vice President Dukakis than by Vice President Quayle? Or George W. Bush by Al Gore rather than Dick Cheney? Or Clinton by either Bush or Dole, rather than Al Gore?
I therefore think that, despite what one may believe about the policies and actions of any given President or election rival, that an oppositional Vice President not only guarantees the party out of power a voice in the Executive Branch, which is certainly deserved in a Republic where the minority retains rights against majority power, but it also keeps the government pulled centerward in an era when demagoguery and partisan politics often tempt leadership to steer the ship of state off a cliff just to prove its leftism, rightism, or whateverism to what it sees as its own constituency. A President needs to be reminded as often as possible that he or she is the President of the entire United States, and nothing could work better to remind a President of this than knowing that his or her chiefest opponent is sleeping in a large house one heartbeat down the block waiting for an opportune time.