Now, prior to reflashing android using the toothpick method, I've tried by clearing the data, cache etc, from recovery, same outcome. I think the bootloader was reinitialized as well, because I needed the toothpick again to boot LE from the sdcard. As far as I'm concerned the box has a probable hardware failure on the EDID stage. What I'm not sure is if flashing with toothpick and sdcard leaves some partitions untouched, which might be corrupted and the cause of the box's behaviour. That's why I wondered if flashing using a PC is better, and can fully restore the unit.
These are all the partitions on my box (Beelink MiniMX III S905X):
dev: size erasesize name
inand01: 400000 80000 "bootloader"
inand02: 4000000 80000 "reserved"
inand03: 20000000 80000 "cache"
inand04: 800000 80000 "env"
inand05: 2000000 80000 "logo"
inand06: 2000000 80000 "recovery"
inand07: 800000 80000 "rsv"
inand08: 800000 80000 "tee"
inand09: 2000000 80000 "crypt"
inand10: 2000000 80000 "misc"
inand11: 20000000 80000 "instaboot"
inand12: 2000000 80000 "boot"
inand13: 40000000 80000 "system"
inand14: 305400000 80000 "data"
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"rsv", "crypt", "misc" and "instaboot" are empty. "cache" and "recovery" are only used by android. "boot", "system" and "data" are overwritten on reflash. "env" holds the bootloader variables. "bootloader" and "logo" are self-explanatory. "tee" is mostly empty except for the string lost+found, so is not the culprit. "reserved" contains a lot of ASCII strings referencing kernel terminology and some binary data, but the name suggests that this partition is not essential.
You could try using the Amlogic USB burning tool to completely reinitialize the NAND, but you would need the proper image for this. You could also ask someone with the same box (I have S905X version) to send you a dd dump of the bootloader partition, if you're feeling adventurous. The MAC address is stored as a bootenv variable, so I believe the bootloader partition does not contain unique information, but I'm not completely sure.