There have been a few US presidents that have been much better than it is likely for a randomly chosen American to be.

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.