How likely is it that LY will ever get to use the SA airspace?
They turned to IATA and UN but I don't know if they can enforce anything
I would say extremely likely, especially once MBS fully takes over the reins.
ETA: Between hoping that LY becomes more customer service oriented, LY joining a global airline alliance, or LY flying over Saudi Arabia (and other Arab and/or Muslim countries such as Indonesia), I would say the latter is a lot more likely than any of the former.