regarding the Obama clip, i'm impressed Obama can talk so clearly about how airBnB works, for someone who lives in a bubble of security and probably never thinks about where he's going to stay when he goes on holiday, he can describe very well the social reputation components of airBnB.
On that very page a picture description reads "U.S. President Barack Obama and aides Carol Browner, David Axelrod and Jon Favreau working on a speech (...)".
I'm not sure if you mean to refer to only American Presidents or national Presidents globally, but there can be a wide range of them, as far as I can tell. Unless you've looked and found that leaders of even the most far-flung of nations are as described?
A lot of the research around the evolution of intelligence suggests that some large part of what we think of as intelligence is inherently intertwined with the ability to socially signal, prevaricate, and negotiate: to assert social dominance of a group through out-thinking rivals and being better at swaying the majority against those rivals. (Thus why sarcasm is considered inherently witty, exactly to the extent that it's subtle enough to trip up some people but not all.)
If you take the dual roles that a President embodies: that of "managed to out-politician every other politician"; and that of "is believed by the technocrats of their party to be capable of negotiating foreign policy, trade agreements, and military conflicts"—you end up with a fair proxy for that particular 'evolutionary' measure of intelligence.
To appeal to a broad segment of the electorate, you need to a.) be able to predict what will be important to a large number of diverse groups and b.) be able to analyze & synthesize that into a platform & message that will bring them over to your side.
The combination of both is a very rare skill that correlates well with intelligence. Note that it requires both emotional & analytical intelligence: you need to understand what motivates people, and you also need to understand the numbers & segmentation well enough to make sure that you're empathizing with the right group of people.
CEO, BTW, requires a similar skillset. I used to think that CEO was a do-nothing role that you got through political favors, and maybe in a couple dysfunctional companies that's still the case. But in a functioning, market-leading company, being CEO requires that you first understand many different perspectives, and that you can synthesize those into a course of action that avoids pissing off any one core constituency too much.
I have heard from a couple friends that George W Bush was smart, just bad in front of a camera. To be honest, though, my own radar from seeing him all those times - from heuristics and deduction - said he was actually not very intelligent in many areas that other presidents were intelligent in. I think his social intelligence was much higher than Gore's and Kerry's, though.
I've heard it hypothesized that Bush had early dementia. If you watch videos of his younger days, he spoke articulately. During his presidency, he kinda stuttered.
