I think you have summed it up really well Ian. I always wonder if sometimes the "let me have a try" method would work where everything else has failed. Unfortunately that can't be done remotely.
I agree when you say, "if the OP can get to the "request for admin PW" stage, the HD must be working to some degree - not totally failed."
I would like sundug to have another try at the process of booting into recovery, it does take a little timing and persistence to get it right, especially with an older device.
https://support.apple.com/en-us/HT201314 this article does state that even if you get the combination of keys and timing correct you may still get an admin login or firmware password screen before accessing the Utilities screen.
If this is the case then the only alternative I can think of would be to use a different boot source but it would be good to eliminate the possibility of a faulty keyboard first if possible. If that is not the problem then a bootable High Sierra Installer on a USB flash drive would be the best option but any Mac OSX, High Sierra or earlier, should work providing its not too old.
Restarting while holding the C key should enable boot disk options then once booted from the installer, Disk Utility can be accessed to erase the Macintosh HD and Reinstall a fresh copy of High Sierra, assuming the HD can take the rewrite without crashing.
After that rebooting should launch the Setup Assistant and the OP can make the device his own.
The frustrating thing about this is that the previous owner should have done all this before passing it on. The instructions for "What to do before selling or giving away a used Mac" are freely available online and if you are the admin it is a simple process that requires no more than a few minutes.