WSJ story on academic study. Gist of story is that China cannot really move into high-income without dramatically improving its government, but the converse logic also holds: we shouldn't expect too much from governments until their society's per-capita income level gets up there. Yes, there are exceptions in each (rich-enough Russia, rich-enough emirates), but the basic pattern is clear enough.
Why the lag? It takes a demanding citizenry to get good governments, and citizens get more demanding, the more money they have. It's really that simple.
At the end of the day, all things being equal, there's no question that democracies outperform autocracies. But the "all things being equal" part doesn't include the catch-up phase, like the one China is going through now.
When does that "catch-up" end and the democratization kick in? The people decide that, usually between $5,000-$10,000 per capita income.
Note that the numbers above are PPP, so high. China measured less relativistically sits at about $4400.