I would prefer a process like Sortition: the selection of political officials as a random sample from a larger pool of candidates -- preferably, all honorably-discharged military veterans of sound mind between the ages of 36 and 68, inclusive.
Imagine getting a phone call in the middle of the night, and the voice on the other end announces that you have been selected as President for the next four years. You have just a few hours to get your personal affairs in order before the motorcade arrives at your door to take you to the White House, where you are sworn in the next day.
All you have to do is veto congressional bills or sign them into law, meet with foreign heads of state, declare emergencies, pardon the occasional convict or two, and look 'Presidential' while doing it.
Four years later, someone else gets selected, and you go back to your old life with the cheers of a nation ringing in your ears and a healthy pension filling your bank account for the rest of your life.
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The mere fact that science may not yet adequately explain an object, event, or experience does not mean the immediate explanation should automatically default to a conspiratorial, extraterrestrial, paranormal, or supernatural cause.