AUGUSTA, Ga.—As Rory McIlroy fell to his knees after winning the last Masters, the player who nearly stole it from him was left standing on the 18th green with his head bowed and his hands in his pockets. For the third time in his career, Justin Rose had fallen just short at Augusta National.  In some ways, it was astonishing that Rose was even in contention. The only man older than him ever to win the Masters was Jack Nicklaus. But nearly three decades after first bursting onto the scene as a teenager, Rose is experiencing an unlikely late-career renaissance…