I've ordered a replacement SCART cable, since I'm beginning to suspect it being the primary fault.
In the meantime, I'm going to attack this thing with a multimeter, see if I can't find this bridge, since my busted-ass eyes can't seem to locate it.
EDIT: I looked over the close-up pictures a little more, and noticed that some of the resistor legs have gold circles around them, and others got filled in with solder from the back of the board. Maybe that's the cause? It's the only thing that's consistent between all three boards. I'll try fixing it and see what happens.
EDIT THE SECOND: Tried everything I could think of. Multimeter shows no bridges. Waiting for the new cable before writing this batch off entirely.
New cable arrived, and I get the same result. I must have screwed the same thing up in all three of them. What that is, I don't know. I can't see it. What I do know is that the pots seem backwards. I turn them to the right, and the picture gets brighter. This is labeled as "-" on the board. Left is "+". I turn it left, and it goes dark. Seems wrong to me, but I don't know a damn thing about any of this.
Guess I'm ordering a new batch at some point and trying again. Or just buying a completed one from someone else.