They treat cultural industries the same way as other industries and with the same economic strategy.
They have long encouraged joint ventures between Hollywood and Chinese studios as a means of studying Hollywood production methodologies and technologies. This is well known and documented. Here's an article from 2012:
世界の政治、経済、社会についての最新ニュースを、豊富なコンテンツと共に多面的に展開。エンタメ、スポーツに加え、世界の文化やライフスタイルについて注目の情報が満載です。
jp.reuters.com
There were major efforts to get Chinese Blockbusters by getting joint productions with Hollywood stars in, for example The Great Wall with Matt Damon, but these generally flopped because you don't automatically get a hit just for having a star.
His role in "The Great Wall," a $150M historical epic, could launch a new era for the country's fledgling film industry
www.cbsnews.com
In recent years, China has started to close its market to Hollywood to try and promote their own films. This is because they had decided they had learned enough from Hollywood and then aimed to reduce domestic competition from Hollywood movies to get more money towards their own movie industry which they believed would translate into global dominance.
Chinese domestic films once again dominated the country's box office amid an annual summer ban on Hollywood titles.
www.latimes.com
Indeed in 2021 the 2nd, 3rd and 6th highest grossing movies were Chinese, even if 99% of their box office takings was domestic:
en.m.wikipedia.org
It's worth noting that technically these movies are competent productions, with production values comparable to those of Hollywood blockbusters, however this doesn't translate into a good or popular movie. But if you were reading Chinese state media at the time there appeared to be genuine expectation that The Battle of Lake Changjin would be an international blockbuster. For instance:
There was also considerable sour grapes when the movie failed to meet expectations. They absolutely do seek to promote Chinese cinema
Police officer Ke Lu from China's Public Security Bureau introduces the methods of film piracy
en.people.cn
Not to mention that China spends billions on their foreign language media (CGTN), albeit to limited success:
Is it worth it?
www.economist.com
The reason they fail is a combination of cultural barriers, political interference limiting creative scope of the movies, and a flawed understanding of what appeals to non-Chinese audiences. It is absolutely not because they aren't interested.
It's also clearly not true that they don't want to export their system. China tends to support authoritarian regimes and has a vested interest in undermining democratic rights because they are perceived as a threat and are less stable partners. This lies behind their alliance with Russia and support for Orban's Hungary.
Beijing’s money and technology is fueling repression worldwide.
www.foreignaffairs.com
They also aim to support the Thai military dictatorship and monarchy against the democratic movement:
China's efforts to influence Thailand's media environment have bolstered the Thai military and monarchy.
www.cfr.org
The collapse of democracy in Cambodia is also a by-product of Chinese influence:
As Cambodia fully embraces the “China model,” the country’s role in the geopolitical competition between the United States and China is becoming increasingly important.
thediplomat.com