I don't think Quyan "Schools" existed. From where are you interpreting this?
My lead in assumption was that Mihtrul was a quyan 'school' as I thought Mimara's mail was made of nimil. Quite mistaken, it is as hard as nimil.
Otherwise there is only the fact that the practitioners of the aporos were 'seduced' by the Inchies and then created chorae, which made me suspect some 'school' type grouping.
So agree with the fact that School is likely an inappropriate term for whatever system of organisation quya had for learning and practicing magic.
OTOH the siqu helped set up the Schools of the north, so perhaps it is likely that they were patterned after nonmen organisations.
The Artisan. The Siqu founder of the School of Contrivers, the Mihtrulic
I do not think we can say sorcerous ability was a prerequisite of rule, and it was certainly not a requisite for Ishroi.
Well, this is my argument but we'll see how it fleshes out. I'm definitely asserting that at some point in Nonmen history, sorcerous ability became a prerequisite for rule.
As noted, sorcerous ability is hereditary amongst nonmen, but rule does not appear so. There are no nonman princes after all.
TTT glossary describes Ishroi as the name given to the nonmen warrior castes. Note plural.
Quya is the generic name for nonmen magi.
Siqu is the term for nonmen involved in the Tutelage.
Which leaves us with one broad term for some warrior castes that could include quya and/or siqu.

Interestingly, there is some discrimination between Ishroi and Quya when RSB wants to make clear that certain individuals are not Quya:
Two Ishroi, renowned for their valour–Misariccas and Runidil–and one Quya… ... Cet’ingira.
More interesting stuff, Gilcunya is described as the
holy tongue of the nonmen quya and is a debased version of Auja-Gilcunni, the base language of Cunuroi - seperate again from Ihrimsu, which the nonmen of Ishterebinth speak.
Culturally, this suggests different languages between mansions.
Can we assume that quya were more generally involved with religious and philosophical matters when they were not also Ishroi?
4 revelations suggests that Ishroi spent most of their time in the field (the curse of Ishroi is to not know that their children are their own).
This would leave little time for religious, administrative and other more productive roles required in a functioning society.
There are more references to Ishroi than Quya in general when we read through the Cuno-Inchoroi wars Neither CC or NJ show any indication of being more than Ishroi. NJ cuts off CC's head in their final encounter, in4Revelations we see CC struck down by a nimil spear. This suggests physical combat rather than sorcerous. So if CC was Quya, then NJ must've held a chorae, in which case NJ could not be Quya. Logically, it appears at least one nonman king was not Quya.
Ciogli breaks Wutteat's neck with his bare hands. Ingalira (Siol Ishroi) strangled Vshikcru (inchie).
NC in 4 revelations is Ishroi but not Quya.
Apologies for the meandering post :p