As to the problem in the OP, my general understanding (not specific to Tesla's mobile connector) is that the EVSE is doing a ground assurance test, in which it intentionally tries to pass a small current from hot to ground. If it finds the ground resistance too high, then it fails the ground assurance test. The high resistance ground typically still shows proper voltage measurements with a multimeter, unless you use one with a "low impedance" mode and check it in that mode.
To test this, if you have an incandescent light bulb and a socket with some pigtail leads, you can try testing the light bulb hot to neutral and compare it hot to ground. With a high resistance ground, the bulb won't light or will light more dimly than hot to neutral. Here the light bulb is just a low impedance tester.
As to the info provided, I didn't study it all carefully, but my questions are: what is the spatial relationship between the panel in the second picture and the panel in the first picture? Are there any intervening panels, or does power go directly from the former to the latter? What is the wiring method between them, i.e. what type of conduit? And in the first picture, the upper left conduit opening in the back, there's 3 black wires emerging from there that connect to the panel main lugs (one of which is taped white near its termination on the neutral bar on the right side); is there a 4th wire coming from that opening, that is somehow fully obscured?
All of the discussion about earthing (ground rods, water pipes, gas pipes, etc) is a side story and not actually relevant to the issue at hand.
Cheers, Wayne