If you are a leader, it means there is something special in you that calls others to follow. Intelligence, charisma, integrity, compassion, courage, these are what mark a leader. While the ability to relate to people, to listen, understand, and identify their concerns, is crucial, I doubt that it makes one an "average person." Sadly, I find those qualities to be rare. Even in good people, there is a tendency to do what is easy, to follow the crowd. There is the attitude of talking more than listening, of instant gratification, of "me first", of never going one step further that I find in the average person. A true leader makes sacrifices the average person does not make. Thus, I do not think a Presidential candidate should market themselves as average. Nor do I think that the American people should value that over all the qualities I've listed above.
Second, I'll admit, I want my President to be "above average" in a few qualifications. I want my President to be intelligent and well-educated. While a Bachelor's or Master's is not the only thing that should qualify you for the Presidency, I want to see that my President cared enough for her or his education to pursue it full throttle. If he/she doesn't care about his/her own education, how will he/she care about the education of the nation? I also want my President to be wise on foreign affairs, to have traveled a bit. As President, you would have to deal with the leaders of nations, all of which have their own diverse cultures, internal issues, and complex histories. While I wouldn't expect my President to be an expert on all the nations of the world, I'd want them to be familiar with world issues and have some foreign experience. No, seeing Russia from your house is not world experience.
"To whom much is given, much will be demanded." To be President, much IS demanded. The Presidency is no average job. It is a job that demands energy, service, sacrifice, foresight, quick decisions, and integrity. Why should we give it to an average person?