The Saturday tournament at the Buenos Aires chess club hosted its usual lineup of players, including an accountant, a college student and schoolchildren. But this time, hunched over the club’s tiny wooden tables with them, was a new entrant: Peter Thiel, the right-wing tech billionaire and Trump donor. Mr. Thiel — who, according to one of the participants, “did not play badly” and came in third — had recently decamped from his homes in Los Angeles and Miami to establish a foothold thousands of