In the hit comedies “Anchorman” and “Talladega Nights,” writer and director Adam McKay played up the sillier parts of American pop culture. He turned to weightier subject matter with 2015’s “The Big Short,” a comic drama about the financial crisis.

“The world took a more severe turn, and I felt like it was time to be a little more obvious,” he says of that shift, which earned him an Oscar for best adapted screenplay.

“Vice,”...