I agree with your criteria of high-level executive experience. Even in my small town of 22,500 residents, the difference of being 1 of 5 on the Township Committee versus the "one/mayor" is shockingly huge. You are the center of everything, every decision, every complaint, every idea, every need. You can't hide (and I liked that when my town was crushed by Sandy - we are not a shore town, but lost 95% of power)). You have to make decisions and live with the consequences.

I would include corporate executive experience as well (to governmental experience). Running something very large and complicated and being successful should be a resume requirement for every future president (something that Obama obviously lacked).