Now that they're visiting my place at the moment, I've been messing around with them.
I haven't tested the power supplies yet, just using my own which I tested on one of my units this morning for a few hours.
The black screen C64 I'm pretty confident has something funky going on in the power-side of the PCB.
The "NO MEMORY AT 0" machine was fine in this respect, displayed the message and dropped me at a READY prompt.
First I checked for a hot DRAM chip, or one that was slightly discouloured, but no physical signs. Since I had the ready prompt, I figured I'd try a mini RAM test program squished in to one line:
FOR I=2060 TO 40000: POKE(I,255): <verison of code to check RAM contents> NEXT
However this really didn't have the predictable results I'd hoped for that could point to a specific chip. I was hoping to see a specific bit being stuck on, but it appeared almost random. As a last resort, I pulled a random 4164 chip out of a dead 8088 motherboard, and started piggybacking.
On the first try
Still needs testing and the chip properly soldered in, but pretty happy to see it start up.