Since the transmission is shared (or at least similar) to many used in that brands vehicles, would a GM dealership have the diagnostic equipment that could read any possibly stored trouble codes from the transmission PCM?

One other thought is whether you've ever noticed the gear indicator lights on the dashboard (the ones that tell you whether you're in Park, Drive, Neutral, etc) lighting up abnormally? If the mode selector switch on the drivers side of the transmission is faulty/electrically shorted/etc, it may simply be an electrical problem you're experiencing instead of a mechanical transmission problem that's preventing the transmission from going into first gear.