What do presidents really do? They are the chief executive, but they do not truly administer the federal government (no one can) although they can set priorities. They make big political decisions. This is not a pejorative - the root word of political is polis (city-states) - political means the affairs of the polis. They work with (wrestle) the other institutions of the government. And they communicate with the American people about what they are doing and reassure us in the face of adversity.
Let's think about it another way.
Imagine you live in NYC and want to move. NYC is an incredibly complicated place to acquire a home. Realtors have an interesting combination of tacit and explicit skills. There are specific legal and financial aspects of home-buying that a realtor must know. There are softer but still crucial skills like negotiating and the psychology of the buyers and sellers. Then there are the contacts, knowing the contractors who can actually get stuff done in a timely manner or the local officials. Then there is area knowledge about neighborhoods. Some of this will be facts and trends, but other aspects will based on feel. It is an odd combination of specific actions, hand-holding, and personal contacts. It is an interesting analogue to being a political leader.
Now, imagine that you are frustrated with NYC realtors and believe they are all in cahoots (one of my favorite words) and working for themselves playing some inside complicated baseball. So you bring a skilled realtor in from someplace else - an outsider. Or you get a smart friend to advise you - an amateur. You trust this friend's judgement, but they are not actually a realtor. These might be good decisions to make (they are similar to the decisions the American people have made in choosing their presidents for the past 40 years).
But, you would want your outsider/amateur realtor to work with an experienced New York realtor. They would still need access to reliable information about the law, the neighborhoods, and - perhaps most crucially - the contacts (the people who could help actually get stuff done). Your outsider/amatuer realtor is still in charge, they are making the decisions for you. But they will need help.
Time and again outsider presidents have found that - whatever their ambitions for "changing Washington" they need people who actually know something about the nuts and bolts of how Congress and the bureaucracy work to get things done. The vice president is one of those key people, along with the chief of staff. An important component of Reagan's early successes as president was having insider Jim Baker as chief of staff. His predecessor Carter initially did not want a chief of staff, but vice president Mondale often filled that gap helping to set priorities and manage relations with Congress. The Clinton White House floundered under the guidance of outsider chief of staff Mack McClarty (to the annoyance of insider VP Al Gore) and righted itself under insider Leon Panetta. The list goes on.
Some light food for thought. Back to punditry...