Brooks,
The simplest answer is that the hydraulic pressure used to power your suspension comes from the GT's hydraulic power steering pump. Unlike most new cars the GT does not use electric power steering as hydraulic power steering provides better feel and provides a source of hydraulic pressure for a couple other systems including the rear wing and suspension. It takes a lot of pressure to compress and lock out the GT springs. That pump only works while the car is running so Track Mode stops working when you turn off the engine for the same reason your power steering stops working, because the hydraulic power steering pump quits working. Cheers.
Chip