Okay, so having read the Demographics article with more detail and care, a few patterns emerge-- including why there is no substantial Achur separatist movement.
Most Achura do indeed remain on their homeworld as corporate subjects of SuVee. These subjects have been assimilated into Caldari society-- they are not Achura living in Caldari cities; they are Caldari. Presumably, they persist as a separate ethnic group mostly because of the Caldari practice of arranged marriages between individuals of similar appearance (the way the Deteis and Civire have remained ethnically and culturally distinct). "Half-breeds" (like almost every Achur PC I know of, for some reason) (no, I'm not an exception) are looked down on.
The Achura living in rural, monastic communities are a relatively small minority of their own bloodline, and largely isolated from the State. These are the Achura ruled over by what seems (by cross-referencing sources) to be an exceptionally disorganized theocracy made up of dozens of sects led by the Elder Visionaries. As a client state, they are considered "protected," but must effectively convert to the Caldari way of life, religion, etc., if they wish to live in the cities, enter corporate employment, etc.
Extremely interestingly, this seems to be the community that produces most Achur capsuleers.
This answers several questions.
First of all, the Achura do not rebel because they haven't got the fading ghost of a prayer of succeeding. SuVee has been on Achura for at least two hundred years, and the "indigenous" Achur people are a minority on their own world. Most of the planet's population are Achura by blood only. They're good little SuVee citizens, functionally Caldari. If the remaining Achura rebel, they stand to lose their protected status and become nonentities-- legal "nonpersons" within the State. This would be tantamount to cultural suicide, entitling SuVee to brush them aside at will or to wipe them out if it saw fit.
Also, ironically, they're also more fully independent of State control than any of us anticipated or appreciated. As a client community of the Caldari, they are expected to mostly keep to themselves, and apparently have been content to do so.
Secondly, references to Achur leaders in various places do, in fact, refer to leaders outside of the megacorporate structure: the Elder Visionaries. Presumably the monasteries are not their only domain, if only because not all rural Achura need live in one. A lot of details about the lives of rural Achura remain unexplored, such as how high the local technological level is, but we can maybe get a few ideas from the "background" notes, because ...
Thirdly, we're all rural Achura now. All of the possible backgrounds for the Achura bloodline find their foundation in some aspect of the highly-inquisitive, intellectual Achur culture-- which is extinct in the cities. The Stargazers look for answers (or at least metaphors) externally, in the heavens; the Monks look within themselves through martial practice, seeking the perfection of their souls; and the Inventors pluck ideas from the mind of the universal consciousness. Individual characters can, of course, claim to be from other backgrounds, and fair play to them. However, it appears that virtually all Achur capsuleers were drawn to the capsule by Achur cultural traits-- and, ironically, adopted the Caldari culture to obtain capsuleer access.
EEEEEEEEE!
It's freaking fascinating! CCP hasn't given us an info dump this good since they wrote the bloodline up for character creation! This, right here, THIS is worth rewriting a character background for.