I’ve had exactly the same problem with my Xbox One S. I’ve tried absolutely everything that was on my hands, even changing the power source for another non-Xbox one (a normal PC power source). And nothing worked.
I did the opening, the cleaning, putting a big fan (ventilator) on top of it, running it without the fan at all, replacing the hard drive, everything… or that is what I’ve though…
… until I saw this post and the message from Ryan Gilbert here above. And voilà!
It seems that the problem is the internal circuits providing the current to the Disk-Drive, because I can clearly see that when the Disk-Drive power cable is disconnected, the Xbox runs fine (to be honest, is still under test), and with it connected, it stops.
It seems that it’s only the power line for the Disk Drive and not for the HDD, although I can’t be sure as I’m still testing it, and I took the chance to change to an SSD which needs less current, so if the problem was there for the HDD maybe it’s no longer for the SSD.
The positive thing is that keeping the SATA cable connected and feeding the drive with an external source (or bridging the connection directly to the main power source) perhaps the problem is solved and we do not lose the Disk-Drive unit. However, I have not yet tested that.