Maybe you can go back to Grant and Jackson. I don't know what sort of experience they had.
What about the head of a federal department or agency? They have a chief of staff, an office of general council, a staff of anywhere between 1,000 to tens of thousands, and they are audited and overseen by the inspector general. They make decisions that affect an entire industry. Set an agenda, establish regulations (don't start on that again, please), meet with lobbyists and politicians, leaders of foreign governments, etc. However, they are appointed, not elected, if that matters. They also may only have experience in just one industry and set of related issues, unlike, say, a governor, but that isn't listed as one of your criteria.
For that matter, what about the CEO of a Fortune 500 company?