The White House Chief of Staff position, and other unelected White House advisors have a lot of power and impact even in the Cheney white house. I've heard Cheney referred to as the COO (Chief Operating Officer) lately (?John Dean I think?) and that seems more in line with Chief of Staff to me.
When the founders made up the post of Vice President it was pretty much strictly to have spare president on hand.
But they also had an issue with the President of the Senate, That role in other Parliaments was not allowed to vote, but if it were filled by a senator, then one of the states would have only half a vote.
So the VP, who would "otherwise lack any employment" was given the senate presidency. Of course in the old days the VP was usually the runner-up from the presidential election, and so couldn't be expected to be in the White house following orders.
This link is interesting :
http://www.senate.gov/artandhistory/history/common/briefing/Vice_President.htm