Well, I found my starting problem, and it may be unique. However, I suggest all GT owners check for this issue the next time they have the rear pan off.
I installed a new battery and charged it overnight before using it, cleaned the ground, etc. Car still did not start, but the solenoid clicked. I removed the rear pan so I could get at the starter and discovered that, when I gently pulled on the power cable to the solenoid, it fell off! The bend at the angle of the fixing tab was completely cracked through! Further examination suggests that the cable, which is firmly cable tied to the frame, had not been given enough freedom to move when the motor rocks on its mounts, so that over time (30,000 miles) the steel right angle fitting on the end of the cable failed at the bend. See the attached photos. The orange line at the failure point is rust, and you can see some darker discoloration (from carrying so much current) where the last remaining point of attachment was. Now I need to find a suitable tab that I can solder or crimp to the cable. Of course this has to be done while the cable is in the car ….
I installed a new battery and charged it overnight before using it, cleaned the ground, etc. Car still did not start, but the solenoid clicked. I removed the rear pan so I could get at the starter and discovered that, when I gently pulled on the power cable to the solenoid, it fell off! The bend at the angle of the fixing tab was completely cracked through! Further examination suggests that the cable, which is firmly cable tied to the frame, had not been given enough freedom to move when the motor rocks on its mounts, so that over time (30,000 miles) the steel right angle fitting on the end of the cable failed at the bend. See the attached photos. The orange line at the failure point is rust, and you can see some darker discoloration (from carrying so much current) where the last remaining point of attachment was. Now I need to find a suitable tab that I can solder or crimp to the cable. Of course this has to be done while the cable is in the car ….