I don't think "do it out of pocket and try and claw the money out of them after the fact" was ever on our list of options. I wouldn't consider that unless the lawyers suggested it. We were looking into going overseas to have to done, which honestly its another big indictment of this system.
The point still stands about how insanely inefficient this whole system is at the macro level. The fact that lawyers are getting involved in routine medical procedures or doctors are suggesting what they know is incorrect treatment is a sign of deep inefficiencies. We aren't an outlier either. It is profitable for the insurance companies to jerk you around, fight you, try and sneak past denials were they can.