Thanks Tom for this space. I wish I knew the right answer to your question. But the fact is I don't.
I know that representative democracy is the best system available to us and that the alternative, dictatorship, even a "benevolent" one is a horrible option doomed to disaster and abuse.
However representative democracy in modern times, is an exercise in numerological strategy and MadMen like advanced marketing. The winner of the election is no more the best man for the job than the best selling laundry detergent is the best at keeping your whites white and your colors vibrant.
The winner of the election is regrettably a product. The result of a carefully exercised strategy dealing with elector counts, political "Momentum" and dollars per voter shift( how many dollars spent in advertising for how many points shift in the polls).
So the question you posed is a very loaded one, because if eligibility is no longer an issue, you redefine the whole thing.
Let me re-state it: What kind of person could do the best job as President if that person did not need to be run through the process that appeals to the lowest common denominator?
Well, I don't know who but I can name some qualities that are no longer needed.
The President would no longer need to look good on Television (Nixon would have beat Kennedy in '60)
The President need not be a politician, not even one bit.
No more flip flops or changes of message from primaries to national elections.
No longer will the President need to pretend he gives two hoots about New Hampshire, all four electoral votes and all.
POTUS need no longer be a jock of any kind, he could be a nerd( and George H.W does not count! He was a fighter pilot for Pete's sake)
Or a good speaker for that matter, as long as he could be persuasive and effective one on one he could be a good President.
Finally Under Tom's rule the President does not need to represent the mainstream prototypical demographic. He could be younger or older and he could be of any and all ethnic backgrounds and any gender.
POTUS does need to represent the nation though, needs to be strong, effective, sincere, compassionate, modern and inclusive. The President of the United States must have a broad vision of the future and have a firm grasp of the past.
I hope this narrows it down a bit.