Maybe people in the 90s assumed "prosperity" in general would be a panacea; but the hypothesis here isn't really to do with GDP more generally (because that can all be generated and captured by the state, as in middle-eastern oil-producing countries) but rather to do with a certain inevitable demographic shift.
China has already started on a shift called industrialization: a massive demographic shift from poverty-class subsistence farmers, to middle-class industrial workers (e.g. construction workers.)
The next shift after industrialization is a mass promotion out of the working-class and into the middle-class (a.k.a. the bourgeoisie: small-business owners/entrepreneurs), where industrial jobs dry up, people mostly live in cities, and everyone goes into business for themselves to serve a specialized role in a city. Accompanying this is, also inevitably, a massive rise in political consciousness, because you need to study the political climate to effectively run a business.
Now, that second shift is far from happening in China yet. Their industrialization phase only just started 10 years ago, with a construction boom analogous to the one that the US went through in the late 1800s. It'll likely be decades more before there are no more Chinese public-infrastructure projects; before the average Chinese citizen is rich enough that everyone turns their nose up at working in the trades; before the average Chinese citizen's life-goal is to be a vlogger or whatever kids in developed countries want to do now.
But, despite that phase being a ways away, it's also seemingly inevitable. We've never seen a country get stuck in the industrialization phase once it's truly begun. It's finite by definition—the same forces that create this phase of growth, push a country through and past it.
The only deciding factor on how long it takes a state to become democratized, in this theory, is how long it takes to reach the industrialization phase. It took China a long, long time (probably because Communism put them at a standstill in accruing the necessary resources and talent to begin having a working class at all; they had to relent and do some top-down state capitalism just to get anywhere at all.)
And many states are stuck lower down the development path, where they'll likely never reach industrialization without outside help. That "help" usually coming in the form of the world deciding to target them as the newest cheap labor outsourcer. (Luckily, once China is post-industrial, it too will be a labor outsourcer, not a labor supplier.)