Yep, it definitely sounds like something is amiss with the AV board. My understanding is that you have two complete machines, except for the ROM. Have you tried swapping the power supplies? The problem could lie with the PS instead of the logic board, especially if the CPU is not getting warm. Those PPC601s seemed to get warm no matter what.
Do you have something in the PDS slot of both logic boards? If there is no video card or expansion board in the PDS slot, then there needs to be a PDS terminator board in the PDS slot.
If neither of the above yields fruit, try (carefully) removing the heat sink from the problem-6100. Examine the CPU to see if it is cracked. I've seen a lot of 6100, 7100 and some 8100s with cracked CPUs...
While you have the heat sink off, clean the old heat sink grease (may be more of a powder by now) with alcohol, being careful not to rub the grease onto the CPU pins. Add a dab (a tiny dab) of fresh heat sink grease and replace the heat sink. A "tiny dab" because you don't want to put so much on there that it runs off the center of the die and out onto the pins of the CPU. I killed a Power Computing Power 120 (8100 clone) that way once.
Oh, while testing, remove all RAM from the machine. There's 8MB on the logic board. That's enough to get a startup bong.
An old memory also says that the 6100 won't generate video at start-up without a good PRAM battery, but I'm not sure whether I'm confusing it with other pizza box models.