One flaw in the theory
You are compressing the concepts "secondary education," "book smarts," "intelligence," and "common sense" into one solid lump. They aren't.
Here's an interesting side point: How many people in national politics have had an actual education in science? How many have a background in engineering? Business? Or anything other than law school or some form of political science? Hell, it might even be refreshing to see a politician with an art degree.
I'm beginning to think that the Ivy League doesn't so much turn out leaders as it turns out people convinced they have a right to be leaders.