So - current behavior seems to be that from an unplugged/unpowered state, I can plug in the machine and hit the power button, and the machine will chime and begin to boot. There's no disk in it, so it comes up to the 1st graphical screen asking you what you want to do.
Very soon after, the machine will lock up - no mouse or keyboard action is recognized. If you then unplug the machine and plug it back in, it won't chime nor power up in any way.
If you unplug the machine and leave it unplugged over night, it will then act as above, starting up and then hanging not long thereafter.
This feels like a power supply/power management issue to me, but it seems odd that you have to leave the machine unpowered to get it to work. What would there be in a power supply or power management circuit that could cause that? Or - could it be that some component is overheating and requires time to cool before it will work again?
Open to any ideas!