It's baaaaaaaaaaaaaaaaaaaaack.
Update, leaving out a lot of smaller bits.
After seeing the system work fine without a graphics card, I RMA'd the card, they replaced the R9 390 with something else a something 580. Same problem, hung almost immediately.
OK, it must be the motherboard. RMA'd the motherboard. Finally the bent pins fixed! I decided to just pay someone to replace it this time and did that today.
Motherboard replaced, system rebooted. Two tests: install OS updates; install graphics card driver.
Test 1: OS updates installed! This is great!
Test 2: hung the system just like before as soon as I ran the AMD auto detect program and it was scanning.
Tried simply running the device manager - trying to look at the driver, hung.
OK, what the hell. What else is there? Graphics Card, Motherboard, software and this is basically a fresh install of Windows 10 (problem happening before and after.)
What does that leave?
The guy had an idea. Have I updated the motherboard drivers? Well, no.
No one anywhere has mentioned that idea. So he ran a program that updates all the drivers.
Tested again. Drivers updated first time ever! Seems to be great!
Hm, oh wait. then a bunch of problems, some hanging, and trying to re-run the AMD installer got new errors. And the system doesn't even show the graphics card in device manager.
So... it's running now, better, but there are still some weird little things - like the driver update program insists the Microsoft default graphics driver is not updated but gets an error updating it.
We'll see how it goes now as I try to run some games that would need the graphics card.
But it's looking possible that after all this the hardware hangs were NOT caused by bent pins or any hardware problem at all.