The Communists largely maintain power though the threat of Its us or chaos. Which before and during the communist rise China was pretty bad off. This is more or less the social contract we make your life better, you give up your rights.
If it becomes clear the central government can't control the economy. We'll likely see a rise in anti-government protests.
I've been to china once, lived with Chinese Nationals for several years, and currently telecommute with partners in China. There is wide spread support for the party. People do understand that the chinese system is unique, and not necessarily the only government system. They simply believe it is better.
Which from a socio-economic outlook it kind of has been for the past ~30 years in terms of industrialization and infrastructure construction. People have lived their entire lives where the idea of questioning the party seemed stupid simply because the party was, and has been right for their whole life. Not because of life or death punishment. Simply looking around and seeing the cities and lights were the only proof needed.
The party ties its sucess to the sucess of the nation. Which is really what all political parties do, lest we forget Bill Clinton's, Its the Economy Stupid. Economic downturn is a crack in that armor. Proof the Party isn't by itself responsible for the economy, proof the party doesn't have absolute control of the up turn. If the party is wrong on the economy, what else could they be wrong on?
Under the Communists, China put a stop to all of this and became a major power again. There are certainly problems in China, but the average person is way better off today than before the Communists took over, and not just because of economic growth.
Now, one can argue that the Communists weren't necessary to make this happen, or that they even impeded progress. (Fighting a civil war and the Japanese at the same time certainly wasn't very helpful.) But the fact remains that they were the ones in charge while this change took place, and they get a lot of credit for that.
Improvements only came when they abandoned communism, and came up with a fig leaf of 'mao was 80% correct'
It is hard to rewrite history with a different set of rules, but I think it's fair to say that given the experience of Hong Kong, China would have been better off without 30-40 years of communist rule.
I am biased towards Hong Kong having visited and loved the place, so it's always the first one I think of.
It may well be fair to say that. But it doesn't change the fact that the Chinese Communist Party presided over a period which took China from the world's punching bag to a strong, independent power. Maybe they made it happen, maybe it happened despite them, maybe it's a mix, but they get a lot of credit from the population for being the ones in control while it happened.
Its the older generations (I assume who got more propaganda growing up) that carry the parties line a bit more firmly.