For China to actually propogate that narrative and sell itself as a counterforce, it would have to advertise itself as a rival military threat to the US and it's allies, but at the same time it has been trying to keep a low profile and let people ignore it's rapid developments or underestimate it, hence not fiercely pursue the PR directly.
Now that everyone has generally caught on and China is being more commonly mentioned as a threat, China may move in that direction.
But also there's a chance perhaps China feels it unnecessary and would rather work directly with the governments itself to build influence and ties, and let the positive ties with the country's national institutions serve as a way to build positive perceptions. Similar to what was done in Pakistan's case.
That being said I really don't think there's anything to worry about when it comes to CCP's competence, that's something they've proved to not only the Chinese but to the world. No doubt they've also thought of something similar or better.