Hi Schlomsi, thanks for looking at this in detail, and validating most of what I found.
In retrospect, I agree that my suggestion of the second contact relating to GPS is probably wrong. I had tried insulating this contact and failed to get a GPS lock, but I've re-tried it and this time it made no difference. It just shows how hard it is to reliably diagnose the problem!
If you test continuity of the various contacts areas on the case with a simple multimeter, they do mostly appear to connect to ground - but that's just a DC measurement. At RF they will behave completely differently. I'm no expert on how these flat on-case antennae work, but they are likely cleverly etched tracks forming a pattern that matches the signal wavelength. If you look at a UHF TV antenna, it's usually a closed loop.
I was interested when you said you'd tested the phone with the back off. I'd wanted to do that, but thought "how do you hold the battery in, and what about the power button?". So I revisited it, and found that a bit of sticky tape will keep the battery holder in place, and plugging in a charge cable will make the phone boot. Excellent! The SIM holder also goes in fine.
To my surprise I got a better GPS signal with the case off than I ever do with it on, and without even touching the antenna contact. It obviously picks up the signal perfectly well directly on the PCB tracks. I then found that just touching the corner contact increased the signal further, confirming that this is indeed the GPS antenna. There was nothing else I could touch that affected the signal.
I like your idea of copper foil. I don't have any so I've ordered some, and will see what I can do when it arrives. My GPS signal has once again been slowly degrading, I think that mesh oxidises over time.