That is a very good question. One that men have asked themselves over millenias. That is why since the birth of Democracy in Greece, we have had tyrants, dictators, emperors, kings and queens, oligarchs, etc.
I would suggest however that in theory, the politicians sell their policies, rather than themselves per se, although they do sell a bit of themselves also. But the way our system works is, you have a philosophy of governance which is conservative, low taxes, traditional values, screw the poor, and the democrats, sort of distribute wealth, keep the companies in check, tax the rich higher, etc. Any one politician in this scheme is not an individual, but rather a representative of a philosophy, or even ideology. So, it is right that the public decides which they prefer at any given time.
However, it would be quite savagery for the public to decide as to who should be guilty of murder, or rape, or conspiracy, or robbery, or the thousands of cases that go through the court. It too would be idiotic for the lay public, with very limited knowledge of the law, to decide as to who should be a judge, a job which requires high intelligence, but also common sense, a sort of moral structure, so taught through law school. Same as it would be idiotic for the public to decide who should be a doctor.
Isn't your comment a general indictment of American democracy in general? Why have an elected congress making policies then?