When playwright-turned-filmmaker Martin McDonagh first conceived of his dark comedy “Three Billboards Outside Ebbing, Missouri,” in which Frances McDormand plays a woman seeking justice for the rape and murder of her teen daughter, he had no idea the movie would come out in an environment rattled by tales of sexual assault by powerful men. Now, McDormand’s expletive-spewing avenger epitomizes the angry feminist reckoning leading up to its release. “I think it’s a great film to be put out in this climate,” the 47-year-old British-Irish director said over coffee in New York. “But it’s not about just rage and pain. It moves on to a more hopeful, human place.”

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The topicality was pure coincidence, but McDonagh will take it. In the weeks following the “Three Billboards” premiere...