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What is public morality?  
The third way.

On one hand, in our modern Western political systems (democracy), morality is a private matter, just like religion. Only the respect of laws is required, and even that, only in form, not in substance.  
This does not work well, because laws are too approximate from the standpoint of modern science, and thus often contradictory to the facts.

On the other hand, in the traditional Chinese system, morality is public in the sense that officials are recruited through civil service examinations based on Confucian culture. The core of Confucius's teachings is a vertical morality: to obey one's superiors (starting with one’s father), and to lead one’s subordinates in the way one would like to be led oneself.  
Throughout the 20th century, confronted with the economic and military superiority of the West, the Chinese political system was in crisis. In 1912, a Western-style republic was adopted, then in 1949 a communist system based on a one-party rule. During both periods, Confucianism was denigrated and rejected, blamed for China's decline. With the paradox of Deng Xiaoping's socialist market economy and the resulting return of economic and military power at the beginning of the 21st century, the regime shifted its goal from mere assimilation to domination, and sought to define a modern Chinese-style governance system by rehabilitating Confucianism, through the new oxymoron of a Confucian one-party state.

On a third hand, in Russia, at the time of the Soviet Union's collapse, Mikhail Gorbachev also sought a third way, but found no basis for it in traditional Russian culture, while Westerners assumed assimilation would occur. This led to a new dark period.

The fact is that the Chinese are right against the West on one point: morality must be part of the political sphere, for a very simple reason —social ambition must be contained.  
Simply, the right way to do this is called strategic scoring, which consists in evaluating individuals' capacity to perform problem-solving, and not Confucianism.  
We have formulated in very few words what the third way is, which China is awkwardly pursuing today and which was aborted in Russia. Indeed, once strategic scoring is adopted, the notion of a one-party system is no longer the central issue. We find here a key point from the book From Capital to Reason: the central issue is not who governs, but how power is exercised: decision-making processes must be framed.

In summary, Western modernity, which produced its superiority in the 19th century, corresponds to the fact that it applies problem-solving in the fields of science and technology.  
Its weakness stems from the fact that it does not apply it in the field of politics, nor even in the hierarchy of private companies.

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Regarding the 20th-century history of China, see the Collège de France lectures by Anne Cheng, 2008–2009.