(One of) The problems with political representatives is that they _want_ to be politicians. Either as a means to an end, or as a mechanism to better serve themselves or others like them. On more rare occasions you find the occasional altruist.
Since there's a selection bias going on, how can the current crop act as representatives of their populace when they aren't _like_ those they claim to represent?
Unfortunately, among recent US presidents, there have also been many who have been much worse than it can be expected for a randomly chosen American to be.
I do not believe that it is possible to make with any certainty an estimate of which president selection system would have been superior, i.e. how the better and worse presidents would have been compensated in the 2 systems.
So judging from the US historical experience, it is impossible to reject the system of a random selection of the president, like for jury members, as clearly inferior to a president voted by the people.
Maybe a combined system would be better, with a set of presidential candidates chosen randomly from all citizens, one of which is elected by popular vote.
This would exclude the people addicted to power who have progressed inside their party through dubious means, becoming the party candidates.
I'd almost certainly prefer a mechanically determined pool (say, all Walmart store general managers as of 2020 who are natural-born citizens, >35, not felons, etc), and then randomly pick the President from that pool. Maybe let people use negative voting to eliminate 10-50% of the pool first.
It is an idea that is usually so quickly dismissed with "what if we pick a total idiot!?" - as if people optimizing for campaigning as opposed to governing couldn't end up being total idiots as well.