Hey there again, @moisu!
Can you please try the HDD in another computer as well and see how it will get recognized there? If it still doesn't pop up in the other system, I'd strongly recommend you use a docking station or an external enclosure with an AC adapter (for the 3.5" HDDs) to check how the HDD would get detected externally via a USB connection as well.
If it does pop up on the other PC, you should probably try
resetting the BIOS on your rig's motherboard. It will get the settings back to factory defaults, so you will need to re-configure all the manual changes you have made in the BIOS (like boot priority, SSD's SATA mode, etc.).
Hope it works. Keep me posted.
SuperSoph_WD