The drive, that has been cruelly rejected by my Plex box, works perfectly as a SATA attached drive in my main desktop PC. It also works perfectly as a USB attached drive in the Plex box. It's brother, an identical (or so I thought) Toshiba x300 8TB, works perfectly in any socket on the Plex box, and any cable.

I've heard of motherboard drive controllers and channels going wrong, but I've never heard of them developing a grudge against a particular device.

If I can't find anything to fix it, I shall attach it as an internal USB 3 device with one of these, or similar. I can snip the power cable and attach it to the 12V & GND pins from a Molex. That
way it will reboot with the rest of the drives.

Probably get it finished just in time for the motherboard to fail altogether. 
 

I've had several drives that ceased working after a while (usually >2-years), when directly attached via sata or pata motherboard plugs, but work fine in usb enclosures.

I had my first BSOD in aaaaages yesterday, by plugging in an SD card that had been corrupted by a bad Raspberry Pi. One corrupted card completely brought down the OS as the disk management service got stuck trying to interpret it and ignored all other requests. 

Had to disable automount in diskpart in order to access the card and delete the partitions, after which the card worked fine again. 

So yeah, bloody PCs are a pain in the hoop.

Yeah, looking around t'interweb, it seems to be a "thing". Thing is, this drive has now passed a ten hour checkdsk scan and an almost as long SMART scan (HDDScan), and it works perfectly in two different SATA slots on a different, newer, PC. As far as I know, it's only in the old PC with it's 11 year old mobo that it can't be seen. But that said, 4 other drives and a DVD drive work fine, in any slot. So there's obviously something about this particular HDD that stops it shaking hands. And it's something that happened over the weekend.

I've stuck on the latest BIOS and chipset drivers, but without any improvement. It's working OK through USB, and there's a card in a PCIE slot giving USB3 with a spare socket, so I'll mount it inside but connect it via USB.

Yeah, that is SOOO annoying and a real issue with Windows: the way one misbehaving process can hang or crash the entire OS, and there's no way to kill it.