Even the current authoritarian Hegemony in which around 0.25% of the population knows the secret, the system is leaky. I don’t think it’s plausible to teach Theurgy to 20-ish% of the population and keep the other classes from figuring it out – unless you try to shift to an incredibly repressive anti-literacy push in which you try to kill off helots, drudges, and the free urban poor who learn to read. That obviously wouldn’t fit with your benevolent vision.
Fair enough! In response, let’s talk about one of my biggest problems with standard fantasy: worlds that have stayed purely medieval-Europe despite widespread magic. A world with magic ought to be significantly different from our own in its social structures, politics, and technologies. Unless the magic is 100% a stand-in for a real-world technology (“I am Watt the steam sorcerer, fuelling my wizardry with the power of coal!”) it shouldn’t leave the world with the same tech/civics tree we see in any given Earth civilization. There should be odd jumps and Earth-anachronisms.
It’s more than fair to be unconvinced by the way I’m imagining Theurgy would have made the gameworld more modern/industrial in some respects while remaining premodern (from our perspective) in others. And I definitely agree with you that in our world, totalitarianism requires high state administrative capacity. A state with premodern state (in)capacities in all areas wouldn’t be able to keep huge cities from boiling over and demanding more rights, even if it could magically conjure bread and banish cholera.
But Theurgy also massively boosts the state’s repressive capacities. Theurge-Kryptasts can hear through walls, “disappear” dissidents, crush riots, and level slums very effectively without relying on the techniques of social organization and administration that our world’s secret police and gendarmes required. And then there’s the fundamental impact on social control that blood magic brings into the picture:
It’s as true of urban revolts as of slave revolts. For all these reasons, I think the world’s magic system makes it feasible to hold large urban populations in long-term oppression, a system I’ve elsewhere called precociously totalitarian, even though the Hegemony overall still has weak state capacity by modern Earth standards. Magicians have a high capacity to inflict totalitarian terror even without a Gestapo or NKVD bureaucracy backing them up.
In our world, the administrative capacities of the modern state were developed painstakingly over centuries by rulers who faced problems of social control and inter-state competition that they couldn’t solve by easier means. Theurgy – as a “technology” initially monopolized by an utterly dominant colonizing empire, and only slowly spreading beyond that empire’s elite – has provided a shortcut for managing many of those problems.
But it’s not adequate even for the Thaumatarchy’s current political challenges – which is why the Hegemony has been trying at great effort to create more modern institutions of taxation, and using the Telones to render its population more “legible”. That’s the main theme of the Irduin section, which also shows how Theurgy can in some ways ease along this modernization push (e.g. flying Theurges make it easier to develop an accurate cadastral map). We’ll see how believable you find it. 
While this is true, the “select few” is (for most purposes) still orders of magnitude bigger than the current Theurge corps. Not everyone has it in them to be a neurosurgeon, but a world where almost everyone knows first aid and there are a thousand times as many General Practictioners is still a very different world from one where only a tiny few can access a doctor. Similarly, setting aside the nuke analogies, a world where everyone has a gun is very different than a world where only a select few do – and the change from one to the other is highly likely to be destabilizing.