The way I imagined it was schools working a bit like barracks/training grounds, turning regular citizens into scribes, that would then go on to garrison administrative buildings and provide various bonuses. But to work they'd require a papyrus/parchment/clay tablets industry, and the scribes might demand pig meat or somesuch.
Rather than educating the general population, I think it's more realistic for the time period to create an upper class of educated nobility. And it makes sense, after all, educating your farmers and miners has poor returns on investment. It's more efficient at that tech level to just have educated overseers.
It might also be easier to code nobility/scribe pops with special demands than to track the education level of regular citizens.