I think the Great Grandfather is probably the human Joseph, as I saw a comment on one of the earlier chapters pointing out that we've never seen "Joseph's" human form. Seeing as the human Joseph instigated the coup, it seems like the most likely option at the moment.
However, it's also possible that the "Shadow Joseph" we know is simply the real Joseph who has achieved unification, much like the rest of the Great Grandfather's elite cohorts.
That said, another similar interpretation is that the Great Grandfather could simply be some sort of figurehead construct—like, say, a "body double" crafted using Joseph's unique soot power. The one thing that's always struck me as odd about the guy's design is that there is no clearly visible head beneath that weird headpiece of his; not even a defined silhouette like all other Shadows...
Neither of these explanations however gets to Kate's most basic question of "
why?". They have a pretty perfect brainwashing system, and on the face of it there is nothing inherently special about the GGF when it comes to brainwashing either. An important thing Anthony has demonstrated this past arc is that the power and system itself is completely neutral, it can be used to create Shadows/Humans loyal to any goal that its "user" desires. Which means if Joseph was behind it, he could just as easily have made himself the leader and the one everyone thought figured it all out, commanding absolute loyalty. An intermediary wouldn't be required.
Now of course there are advantages to being "the real power behind the throne" with an under control figurehead, since the figurehead can in turn attract any arrows and as a very high up but still "mere" adult Joseph may have much more freedom of action. Pretend loyalty can itself be a shield, because nobody will be expecting a "fully under control brainwashed adult shadow" to be doing anything against the House, and he won't stand out particularly vs the other three in the 3rd floor nor even that much vs the now large numbers of adults. Still though, it leaves the question of why they've been relatively passive and experimental.
The best hypothesis I can come up with for now is that ultimately this all dates back to something in the Mirror's distant history,
before they came to the Island at all. Back in chapter 198, when Anthony recounts his own knowledge of history, the first page feels like it might be an important one:
"The Mirror family was in decline. Our brothers were lost to war, and our lands taken from us. The only thing we had left was an island, afflicted by rumors of troublesome fairies. But to me, that island was my one and only hope."
That line of course was the intro to his father Alfred's goals and stories. But what war, what brothers, what lands? Clearly the Mirrors house was some sort of nobility and potentially somewhat big deal (if an entire island that large and resource rich was merely a minor not particularly desirable portion of their territories), but whatever the war was their side was losing. Yet there must be major powers still out there.
So I wonder if this is something of a Claymore situation: the island is serving as an experimental testing ground for developing a new kind of soldier/power. Joseph could be acting on his own for vengeance, in concert with others for vengeance, or at the behest or threats of a major mainland power. A variety of those could explain why he'd want to make sure that his own personal role was obscure. While within the island they're now very powerful, it may be there are other powers out there that could still crush them. Or Joseph may have remaining ties that could be used against him. But both the motivation and care with stealth and slow/steady development, the major emphasis on experimentation and development vs mass production, a lot would come together if this is part of a long term larger great war. We'll see though!