Does the battery appear to be charged (tries to boot with the power adapter disconnected?)
Is there any difference if you try a boot without the power adapter connected? (fan - no fan?)
Try disconnecting the battery, and boot with only the power adapter connected.
Is there any difference, power adapter-only, or battery-only?
If the fan always kicks up right before the shutoff, you may have a sensor fault. Too bad that the diagnostic appears to not report a failure, but it's not a 100% proof that nothing is wrong. Keep in mind that test or no test, you still can't boot successfully.
I'm not too clear about one thing that you tried. Did you actually try physically removing the internal drive, then try a boot to your external (no internal drive installed)
I'm suspicious that the previous owner may have tried an interesting "mod", or perhaps there is liquid damage on the logic board? Did you give the internals a close inspection, looking for either staining, or even corrosion anywhere on the logic board?