I don't think that I'm willing to give up on the idea that capitalism and liberal democracies do have some unifying link. China will be a test case for the theory, but an argument about authoritarian capitalist systems is that they generally devolve into illiberal oligarchies, which is the case with Russia since the fall of the Soviet Union.
Democracy ultimately provides a check on illiberal authoritarian capitalism which is why, generally, liberal democracy and capitalism have thrived together. It's very easy to talk about "China's success," and look at it from pure numbers, hundreds of millions of people who have risen out of poverty over the last 20 years of opening up their economic system. But, hundreds of millions more still don't have access to basic human necessities like clean drinking water. So, you can say "Well, CHina is the most industrialized nation in the world now by output -- that is success!" but then a challenge to that is that more than half of Chinese do not have access to clean drinking water, and it's not just in rural areas as is the case with no access to medicine, but cities as well (
Guardian Summary of Greenpeace study).
So, how does this come back to democracy? Well, typically, societies that are both liberal democracies
and capitalist economic systems provide a balance on how the entire society grows at approximately the same clip.
Now, for sure, you'll have long periods where there is a rising gap between rich and poor, as has been happening in many Western countries for the last 2-3 decades, but for the most part, those have historically balanced out within democracies, where in authoritarian dictatorships, plutocracies generally develop, and you end up formalizing a permanent wealthy class and permanent vast impoverished class.
Now, of course, there are going to be cases where capitalist dictatorships can develop and be successful, but eventually one dam bursts, either the economic order collapses or the dictatorship collapses. An example is Pinochet's Chile, which in the 1970s and 80s implemented a free market capitalist system and it caused Chile to outpace most other Latin and South American countries well into the 90s... but eventually, Democracy won out: Pinochet's regime is remembered now for human rights abuses and corruption, while his party was swept out of power and he was charged with numerous crimes. Economic inequality grew because Chile under Pinochet was not a liberal democratic society, there were sham elections and it contributed to a growing plutocracy, before his government finally collapsed under overwhelming public opposition. There are other examples as well, but Pinochet and Chile are one of the better, small-scale micro examples of the threat that capitalism provides in a non-democratic system.
