It's difficult (at best) to incorporate the Adamant Caste into the other five; suffice it to say that the big reveal of the Sixth Caste would destabilize Autochthonian religion (such as it is) and morale would plummet into a death spiral from which it would never recover. It's one thing to think that the bogeymen are in the shadows, watching you and judging you; to actually know that the bogeymen are out there is far worse.
To some extent, Autochthonian society already has this, rendering the Operatives redundant. Moonsilver and Soulsteel Castes watch for thought crimes against state and god. They stalk, hidden amongst the masses, destroying proscribed technologies and methods of thought that would injure the Eight Nations. All of the social castes work harder, knowing the Exalted are in their midst, but this only stops malignant or negligent behavior. It is mostly bereft of the kind of necessary sociological self-examination, of the judgment of a society that comes from living outside it. The two castes may only correct elements of the society that go wrong, not if society itself does.
The Adamant Caste does those tasks as well, but they a) also pull double-duty in watching the Exalted and b) specifically watch for proscribed technologies or methods of thought that would injure society as a whole or Autochthon himself. In a perfect world, neither would be necessary, but the Adamant Caste is more a function of Autocthon's paranoia than anything else. In order to be a proper judge of a society, one must exist outside it, theoretically immune to the temptations that even the Exalted suffer from. The Hands of the Ministers have their own society, it's true, but one that is limited and irrelevant to the standpoint of the Eight Nations. This is the thinking of Autochthon, of the Ministers.
I suspect that Operatives saw a lot more action in this regard during the first millennium of Autochthonian society; in the modern day, the societies are relatively stable, injurious elements arise rarely, and most of their attention is devoted to fighting the gremlin hordes. They're probably the most effective operatives (heh) against deep Void Lord intrusion into the societies, whereas the five Castes are more effective at fighting Gremlin Syndrome spawned at home. Perhaps they are kicking themselves over allowing society to become wasteful or their failures in preventing resource-draining wars. They're undoubtedly quite divided over Project: Razor and it's sister projects in other nations.
But that doesn't answer your question, except in oblique terms. They're more meant to be played off in their own private games, and to act as NPCs in your standard Alchemical game. However, there's nothing stopping an Adamant PC from entering into a game, so long as you examine the repercussions of the Alchemical's Caste becoming public knowledge (riots, questioning, Void Lord attention and such) and as long as the PC accepts the fact that every time their anima flares, the rest of the Assembly probably won't give them the gratitude they deserve. That might seem like a small thing, but it's not - there's an innate desire to be held up as a Champion within the Alchemical Exaltation, no matter how humble they are. Humility is a virtue, and there is no virtue without adversity; temptation lies within.
Alternately, a PC Adamant could either make use of artifact transmogrifying Charms and disguise themselves as a highly-competent mortal attache of the other PCs. Or, they could reveal their identity to the group, promising to act as a guardian angel but being cautious not to reveal anything outside the Assembly.
Sidereals work because there are 5 castes of them; they can play their own game and be quite successful. They don't HAVE to be part of a game of Solars or Lunars or Dragonblooded to feel like a complete set. Adamants…feel like an incomplete set. :/
Mostly, it's because their existence disrupts the status quo you see in 1e Autochthonians. Don't blame me; blame the 1e writers. They are sort of their own incomplete set, and they don't map on to any of Creation's Chosen. I even noted this in the ST Chapter, under Five by Five.
There are things you can do about this, and introducing the Sixth Caste to Autochthonian society is a ready-made plothook.
More to the point, the Adamant Caste allows you to examine games and themes of heroism a bit more bluntly than standard Alchemical play allows. The Alchemicals are heroes to their society, and the standard they hold up is that of the virtue of a a thankless, tireless Champion. What happens when they literally don't get thanked? What happens when nobody really ever sees your toil and effort?
The Alchemicals are champions, but their efforts come with a lot of perks. The Adamant Caste shows that Alchemicals are Champions to their core, with or without added benefit. Additionally, if you make contact with Creation, they can serve as pseudo-Sidereals, keeping the Champions and the Exalted Host in line. After all, it's what they do.