The Vice President has a large collection of unofficial duties – exactly which, depend on the personalities and skill levels of the President and the Vice President and how they interact. But they occupy a great deal of his “working” time.
For Eisenhower, Nixon was two things: (A) the “relief President” for ceremonial appearances, particularly overseas, giving Ike, who was in good physical shape (except for his heart attacks) but aging, some break from the incessant need to be somewhere for some special event; and (B) the “stalking horse” and “bad cop.” If the Administration needed to test a controversial proposal, or to “slap a supporter’s hand,” Nixon, supposedly acting on his own, would propose the controversial step or make the public criticism – which Ike could then distance himself from if circumstances called for it. And both men, who knew how to play politics with the best of them, were comfortable with this arrangement.
Johnson was too much a maverick to either do this for Kennedy or to have Humphrey do it for him, but Nixon, President himself, used Agnew in very much the same role. Reagan depended on GHWB’s expertise in foreign affairs quite extensively, and Al Gore’s longstanding Congressional connections made him equally useful to Clinton – as did Alben Barkley’s when he was Truman’s VP. (I’m skipping several VPs whose roles if any I’m not familiar with – which is why this hopskotches as much as it does.)
Mr. Cheney’s extensive expertise and contacts throughout the public and private sectors, and strong policy views, have made him seem to be an eminence gris in GWB’s Administration – with the caveat that he at least was elected to the job, so his influence is not the murky Colonel House style backroom manipulations of the past.
But in general, having a man who can carry part of the burden of the Presidency and whose public role precludes him from criticism of the Administration has been invaluable to Presidents, and those more-or-less-behind-the-scenes capacities are the important, albeit informal, part of the Vice President’s job.
Only on those rare occasions where a tie-breaking vote or a ruling on Senate Rules that has substantive consequences is called for, does the formal element of the job, the presiding over the Senate, become significant. Hence the custom of having the President Pro Tem or another senator occupy the chair the majority of the time frees the VP for his “real work.”