KANSAS CITY, Mo. — The stories passed down about the Kansas City Chiefs’ playoff malfunctions might sound apocryphal, as if embellished over a game of telephone, but they’re not. Really.

The fans who jammed Arrowhead Stadium on Saturday can recite detail after grisly detail. So can those who roamed Kansas City’s sideline: Many of them added to the misery last year, two years ago, three years ago, five.

One player changed the calculus. His ascension terminated an era of despair. His quarterbacking brilliance inoculated the fan base. In his grand playoff unveiling, Patrick Lavon Mahomes II purged a quarter-century of Arrowhead futility.

After every first down, every score, every defensive stop, fans in the upper deck flung snowballs in glee. They kept coming, much like the top-seeded Chiefs, who blitzed their postseason nemesis, the Indianapolis Colts, 31-13, and advanced to host the A.F.C. Championship next Sunday against the Los Angeles Chargers or the New England Patriots.