A friendly game of Go in the picturesque city of Wuzhen this week has given Google an opening to return to China’s good graces, seven years after it left the country over government censorship of its namesake search engine and hacks of its Gmail service.

Few companies can match Google for its global reach, with more than a billion customers each for seven different products. Yet China manages without it, leaving a gaping hole in the global map for the Silicon Valley giant and its parent, Alphabet Inc., as they expand into...