WESTLAKE VILLAGE, Calif.—While in New York to accept the National League MVP award in the winter following the 2018 season, Christian Yelich chose to make an unusual detour: He visited the Major League Baseball headquarters in Manhattan to meet with the newly reorganized marketing department.

Just six months earlier, commissioner Rob Manfred created a bit of controversy when he appeared to blame Mike Trout, the sport’s best player, for his relatively low public profile off the field. “Player marketing requires one thing...