So basically, you call this NYT article BS, which I agree.
The only time one would complain about "Power" (hard, soft, sharp, smart, etc.) is when one doesn't have it, or face the erosion of its own.
What the US, and Western allies, had back during the Cold War was the most insidious influence -- culture.
Here is the definition of 'insidious'...
proceeding in a gradual, subtle way, but with harmful effects.
It is 'harmful' only to the target. Not that it is harmful in its nature. In other words, Western values are harmful to what the communists wanted to preserve under their rule.
When I was a tourist in communist East Berlin, my Levi's denim jean, US currency, Mickey Mouse, or the US Constitution, all were considered 'harmful' and threatening to East Germany. Threatening because they were desirable by the populace.
Culture is soft power at the most atomic level of any society -- the individual.
Above the individual is the community and over all is the state ( government ). We won the Cold War at the atomic level.
You can destroy a tank two ways: With a weapon or with water.
With a bomb or a missile, you can put the tank out of commission for a long time. If you put the tank in water, eventually, the water will literally dissolve the tank. It just takes time.
We won the Cold War because we won over the individuals. Just like water, it took time but the Cold War was won with practically no bloodshed, at least not in the context of warfare as we know it. The Western values were insidious to what the communists wanted.
China's type of soft power is effective at the state level. Like it or not, around the world, people want to be like an American or a European, not like a Chinese.