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Emperor
Actually, the only order is the era they come from. For example, in all games you'll find Sagan, Kwolek and Abdul Salam as the 3 Great Scientists from the Information Age, but the order they appear is quite random. I've seen all 3 come first in a game or another.
If we're talking about a pool of GP, in my opinion they should still be separated by era. For example, 5 Great Engineers in the Industrial Era, from which 3 could be recruited, at the choice of the current recruiter. It doesn't make sense to recruit Sagan way back in the Ancient Era.
Another problem about tying GP to governments is the same that occurs with some policies: some won't be recruited at all. I'd never, for example, recruit a GP that enhances Monarchy, unless it was something in the order of 2 extra policy slots to make Monarchy OP.
I just finished a game, and Sagan never appeared (I was watching for him) I did manage to snag Stephanie Kwolek, and that was enough to put me over the top for a science victory. I had 3 high-production cities and one spaceport. My capital built all the space projects, another city built bombers and fighters, and the other pumped out tanks. Japan was right on the verge of winning and I came from behind for the victory. Gandhi took Wernher von Braun; if Hojo and gotten him, Japan probably would have won.
But what really brought me to this thread was to ask if anyone uses great people as scouts? I played a game as Russia recently, and very early on I was drowning in great artists and musicians. Since they are immortals now, they might be good for scouting out hostile territory. I did use one to clear a barb encampment