That's good reading! But it looks like the solution the two of you came up with was to boot into fastboot from diags mode, and diags mode isn't working for me. So I think I will try getting linux running in a virtual machine to run the fastboot tool, unless someone can get the PC fastboot tool working.
I wonder if things would change if I could charge it all the way. It only seems to want to charge when it's in "plug-me-in" mode - no matter what state it's in with a black screen, and whether wall-charger or computer charging, it always seems close to running out of batteries. Of course this would be easier to investigate if I had a way of querying the battery state - very hard to infer from the limited information given by its behaviour.