In some ways, economic dominance leads to cultural dominance. America, Europe, Japan became first world countries earlier than the rest.
I personally think China itself gave up on exporting culture to the West. I see Tencent, the dominant player in China's gaming market, would rather buy Epic, Riot, Supercell than making games for the West. I see singing contest in China inviting South Korea and Kazakhstan artists to compete and none from the West. In the movie scene, it wasn't too long ago that Wanda bought Legendary Pictures too.
When it comes to strictly entertainment, they understand that it's futile to go against decades of conditioning of your average westerners. For their stuff to be a mainstream hit, they would have to western-fied everything, including its aesthetics. The last few surviving triple A Japanese games all have white leads but the Japanese DNA within the game is a hindrance. And if you go 100% western, your stuff would be losing your identity.
Hence, it's so much easier to just buy over everything.
China has only rise like for the past 20 years, they have a very strong regional focus first. I am seeing more and more China shows in my country, Singapore. Their equivalent of The Voice even went as far as to scout South East Asia contestants who can sing in Mandarin.
For me I see it in this manner. Because they are such a huge market, they act like a vaccum and sucks everyone nearby into it. It's kinda like Hollywood where Europeans are heading towards there to work then staring local films, regardless of how excellent these films could be.
TL
R - Soft power takes time and China quite frankly can't be assed to export their stuff to the West. They rather put their money and effort on regional countries.