Can Chinese culture compete with centuries of western influence? As the article mentioned language is probably the biggest barrier to spreading influence, but are there other ways? Have they taken any action to export values that are in contrast to western culture? If not, are there values that would be beneficial in increasing Chinese soft power?
It's not the language barrier that's keeping Chinese culture / soft power from spreading, it's FREEDOM.
That's what makes America's soft power the strongest in the world, and it's why China will never get to be a serious superpower without it.
For real culture to exist, there has to be real freedom of expression, not just sterile rehashing over ancient techniques.
For great art to exist, there has to be tolerance of disharmony. If the art / music / fashion / culture of the world of the last century or so has proven anything, it's that there needs to be a hearty dollop of disharmony not only tolerated, but encouraged.
If your country is so hell-bent on maintaining "harmony" at all costs that it's using members of peacefully dissenting religious sects as organ donors and forcing abortions and sterilizations on people so they can't even exercise the most basic of human freedoms, well, you might as well stick to drawing pandas and doing calligraphy, because your official culture doesn't have anything of value to offer the world.
I know there are lots of rationalizations for this kind of behavior, but art is an outgrowth of freedom. No freedom, no art, except in rebellion.
In a free society, ideas are shared, evaluated, discarded, remolded, so that classical ideas and transformative new ideas can have free interplay, to create ongoing combinations and improvements.
In a society where those ideas are not freely expressed, they are stunted, and either collapse under the weight of officialdom, or build and grow to the point that the bureaucracy is overthrown and freedom reasserts itself, neither of which are conducive to building official soft power.
The Chinese people are HUGELY capable, but if one of their great artists is prevented from leaving the country to attend the award ceremony for one of their Nobel Peace Prize winners (who just happens to be imprisoned for the advocacy that got him the prize), then it's pretty clear that the Chinese government is too fearful of the capacities of its own people to let them develop the sort of culture that could make them a world soft power leader.
We all love pandas and calligraphy and Chinese food, but all of that with a big side order of INDIVIDUAL FREEDOM would really make China a force to be reckoned with.