Yes, it might be considered that the game was invented in China, because a Chinese professor Ling Hongling claims that the game originates from an ancient Chinese game, which later in 12th and 13th centuries was brought to Europe by Mongols.
You don't have to invent something to be good at it. If we use your logic it would mean that only people who invent something can do it or use it. With this in mind that would mean that only the Scottish could be good at golf since golf came from Scotland.
Golf
The Chinese did invent the toothbrush. They invented it in the 15th century.
The Chinese did not invent space travel or the automobile. The Chinese did not invent pizza or chocolate chip cookies or ice cream.
Yes
in 1824
Yes, the Chinese did invent porcelain. It was invented by a man named Tao-Yue.
Yes. Chinese did invent the clock
The Chinese did not invent the Tank Atomic Bomb Airplane Printing Press Helicopter Car Submarine
yes
no, they cultivated it
Yes they did.