You're not being ignorant; it's a pretty abstract line of reasoning, I'll admit.
The logic (again, IMO) is that if large swaths of these governments can be superseded by AI itself—which they're abundantly aware of—then over time we'll see their size organically shrink (but I would anticipate that shrinking to be preceded by an explosive growth first—just like an animal will puff itself up as a defense mechanism).
The money to labs is another fear/panic move: fork over money to your destructor so that it's more benevolent towards your favor when it finally brings down the hammer—that's why they tried to assert dominance over Mythos, etc (appear strong when you're weak). That may sound crazy to someone technical, but realize, most of the people coming into contact with this stuff think it's fucking HAL 9000.
Assuming that foreign competition makes the frontier labs irrelevant (based on my own recent experiments w/ stuff like Kimi K3, this is increasingly likely) or at least, not worthy of current domestic investment levels, you'll see a pullback in the markets. The markets are the only gunpowder the current day government has for propping up their illusion. If that goes, psychologically, it will be impossible to convince most citizens short of direct violence that they need to remain in power much longer.
I'm not presenting this as a "tomorrow" or "ten years from now" shift. I'd guess this will be more on order of 30-50 years, stair stepped. And in the step down, I anticipate full bore authoritarian chaos as the candy and toys get taken away. Maybe even a dictator or two emerging in the places we'd least expect.
So, I would read this less as "reading my crystal ball" and more like "some internet guy's moderately schizophrenic read on the current state of chaos."