Weiqi is recognized by Asian sports organizations because its development is concentrated in Asia, so it will appear in the Asian Games.
As for why chess doesn't appear in the Olympic Games, just like China's martial arts, the recognized events are not mandatory, so generally speaking, if the host country doesn't choose, it won't appear in the Olympic Games. I remember that once the Asian Games seemed to be held in Qatar, and that Asian Games chose chess as the special event of the host country. At that time, Zhu Chen, the former chess queen of China, and her husband, Chess King, were sitting in the game.