FWIW..unless the Westerlies were really adverse I'd still expect the BA16, even on a 777, even in winter, to route via Pakistan and Afganistan ( and of course go any further North and there are terrain issues, even on the 744).
However as parabellum has alluded to there is the slot system ("BOBCAT") that regulates flow into Afgan airspace at peak times - I wonder if the OP's flight had a slot time problem and it was decided to route via Saudi to avoid a delay? I've certainly seen that tactic used
on that flight in the past.