Um … I’m sure you know that antistatic foam is conductive. That’s what makes it antistatic. Sure, not as conductive as metal, but putting random parasitic loads and low-medium impedance shorts all across the back of the motherboard would absolutely not my first choice in a debugging technique (or anything else).
I wouldn’t have guessed it would start at all. I guess I learn something new every day.
You say it came in with a cooked power supply. While PSs can cook themselves over time, some oddball positional short could, too. However…
I’m wondering if it is strictly positional. For example some PCs have an mobo light sensor to protect against tampering (activated as a BIOS option). Tilting may expose it directly to some ambient light source. I saw behavior like that while repairing VCRs, due to the IR LEDs/phototransistors commonly used to detect (e.g.) tape and roller position. Did you try it with the case cover entirely closed? Entirely open?
Did you check that the Li coin cell was clipped in properly? BIOS batteries can cause funky symptoms, esp. during boot, and some holders are capable of partially clipping. I’d imagine you covered that when you reseated everything, but I had to ask.
I saw something like this from a free screw inside in a CD drive I fixed, but I don’t recall the exact symptoms. I’m sure you tried it stripped down, though.
I once had to remove some grit (glass? rock?) from the bottom of a RAM slot with a dental pick (not just one particle, either) to get a computer reliable, but I know you tried reseating, but I didn’t notice in several reseatings before it finally felt wrong . I don’t know how it got there, but it wouldn’t blow out with compressed air. (I promptly cleaned my bench)
I can see PCB microfractures doing this under the weight of some heatsinks, but that’s a mobo problem requiring replacement. I can see a broken wire in the mobo wiring harness doing it to, but I’ve never seen that actually happen.
Sorry, just some oddball ideas for an oddball problem.