How did the Chinese religion reinforce the centrality of the family in Chinese social life?

The Chinese encompass a huge number of people living across millions of acres of land in mountainous, desert, coastal, and tropical regions. Their religion isn't easy to boil down, simply because the variety of Chinese lifestyles—even in an empire with relative homogeneity—does play a part in the variety of religious beliefs and practices.
Traditionally, the Chinese people have practiced Buddhism, Confucianism, and Taoism, with an underlying and pervasive element of nature/pagan/folk religion. This last element is the type of religion that was practiced worldwide long before "organized" single-god religions, such as Buddhism or Christianity, flourished. Examples of "folk" religion include the Celtic Druids of Europe or the Brujas (witch doctors) of the South and Central Americas.
In China, folk religion included superstitions; a powerful, mysterious natural world; spirits of the dead; clan spirits; and Feng Shui. These beliefs continue to mix with whatever "modern" religion is practiced in contemporary society.
An analogy is the way Mexican and Central American people practice Catholicism. The Catholic saints get mixed up with nature, witchcraft, and family, and animal totems get blended into Catholicism. Hence, the Day of the Dead is a ceremony that celebrates the power of dead ancestors, even though the Catholic religion proposes that these people live on in an afterlife.
Because of the strong influence of folk religion in China, the idea that ancestors are watching over their descendants remains. Although Buddhism does not propose any family or clan spirits, practicing Buddhists often still visit the graves of their ancestors and may believe they can be "haunted" by their ancestors and whichever family demons have been left unresolved.
Although Confucianism and Taoism stress balance and harmony in life (the latter with more of a slant toward balance with the natural world) and eschew superstitions, a Chinese person practicing either of these "modern" (post-500 BCE) religions may still use Feng Shui, may still venerate ancestors, and may still believe in spirits.
The bottom line is that the folk wisdom of venerating ancestors has been incorporated into modern religions. The importance of family and clan to survival became a "religion," because learning from older family members (e.g., grandparents) was a practical reality and because the belief that family members "lived on" reinforced the secure social value of being part of a lineage. Ancestor worship acknowledges the wisdom and knowledge of those who came before.
The veneration of ancestors by the Chinese is far more important than in Western culture; it is one way in which the Chinese (and many other Asian cultures) put the group ahead of the individual. The group must be cohesive; group rules must be followed, and individual wants and needs are far less important. In ancestor worship, the importance of the family is extended beyond the immediate family to all generations that came before. Thus, the individual is part of a much larger and more important group and views themselves as having a role to play in the group, often one in which they have little individual influence.