China's motivations are it's either a social thing where they believe they're doing it for the betterment of society or they're in it to get a piece of the pie.
Example We Chat, could have easily used Whatsapp or Line or any other messenger but makes their own. Google -> Baidu, Twitter -> Sina Weibo. Etc and the list goes on.
I'm surprised they don't have their own steam client yet, or maybe they do and I'm not really aware of it. Banning PUBG would mean Chinese players and Chinese money stays in China by buying other Battle Royals games. The alternative option would be PUBG gets published by a Chinese company so that China gets a slice of the profit.