OK, so if I read correctly, the Sony HU works fine in your other car, your factory HU works fine in your car, but the Sony refuses to work (except the motorised face) in your car. Right?
That odd, you've taken the troubleshooting steps to narrow it down and have essentially ruled everything out. It seems though that possibly the existing head unit is not using a wire that the Sony does, or that it's making a better connection with the harnass.
First off, I know many new head units need to be "juiced up" for a few minutes, that is they have to have power supplied to them for upto 2 minutes before functioning properly. Make sure you are leaving it plugged in long enough. Second, the only wires that should matter initially are the yellow, red and ground. Get a volt meter and test yellow and ground and red and ground with the car on accessorry. See if you get readings on both. No reason why you shouldn't since your old head unit works.
Check to see if possibly the any pin in the harness or on the back of the stereo is lose or charred from the mishap.
Last but not least, I had a sony wiring harnass fail on me, and it turned out that one of the pins in the sony harness had gotten pushed in or too far to make contact with the deck. It was the yellow wire in my scenario. I had to get a small screw driver and push it back into the harnass and then zip tie all the wires close to harnass to keep it from popping out again.
Hope this helps.