I'd be okay with the separate ceremonial head of state being determined by game show. Hell, American Idol's multi-round elimination system is already a better democratic structure than most democracies have, go for it. (The ceremonial portion being who decorates the White House Christmas tree, has tea with the Queen Of England, pins medals on people, that sort of thing. If they're not a complete clown maybe they're also an ambassador.)

I don't have any strong opinions on whether actually doing this is worth anyone's time.