How is some bit of software going to help if the drive can't stay on long enough to be recognized by the (BIOS?) and therefore unable to be used?
The problem happens just as much, read: always, on my 450W bench supply.
Is there any way to encourage the head to sit still in a different spot or something so that it can move properly?
Ok, If you get the drive going (as you did in the video) - then carefully place the top cover onto it with two screws in opposite corners, then see if it boots OR if you can install a system onto it. Yes, this means NOT turning it off once you get it to calibrate (as it did in the video), OR if it does spin down due to a reboot, you need to nudge it again.
The idea with the exerciser is that any lubricant on the head armature pivot point or on the disk platters is JUST enough over the gram-force the magnetic head actuator can provide. If we exercise the drive under warm conditions, it might be enough to displace/re-distribute the marginally hardened lubricant.
When these drives would do this it was usually the first cold day of the onset of winter. At that point the drives were about maybe 8 to 10 years old. That drive is now MUCH older than that.
Back then, we had plenty of alternatives to replace SCSI drives, so it wasn't something we delved too deeply into, but we believed we'd worked out WHY it happens. Fast forward to today, and drives are no longer cheap or available new, so I'm simply making an educated guess as to what might work.
I'm quite happy for you to search out a replacement drive, and that's what we used to do, but times have changed. The problem was fairly common on these quantums.
The next test will be to see if the drive functions (without powering off or rebooting) after giving it a nudge to get going. In my experience, once they calibrated, they'd work fine until power was removed.
EDIT: oh, and to answer your question (whoops), there isn't really any safe place for the heads to "sit" when the drive isn't running, other than where the head lock places them (the 'landing zone').