I do not have experience with repairing the specific board you show. But some comments about your photos:
Do the 2 black wires that plug to the circled pins in the photo above come from the track (one from center roller, other from wheels)? That is, they seem to be co-located with a bridge rectifier which suggests those 2 pins are incoming AC. That said, could the 2 wires be swapped? Or did you try both Whistle and Bell to activate the car? If wires are swapped Whistle would be the trigger.
That's a cube relay in the middle of the board. Can you hear it "click" when triggering? If it is clicking but no action then the board is probably good and it's more likely an issue in the wiring to the mechanism or mechanism itself. Again, I'm not familiar with the Signalman mechanism but in the MTH operating reefer, there's a lever-leaf "limit" switch that stops the action after a cycle of man swinging out and back. If this car is designed like that, the limit-switch might have mechanically shifted such that it is always indicating that the cycle is done thereby cutting off any action. I posted a video of the limit-switch in action (albeit for an MTH operating reefer) in this OGR thread; if this is where the hunt takes you, open your working caboose, find the limit-switch and observe it's behavior relative to the broken car.
I figure you probably don't want to take the chance of swapping the board in your working car...but that's obviously a way to see if the problem follows the board or the car's mechanism.
Finally, it appears you have alligator cables. I'd think if the two circled pins I mentioned earlier are indeed track voltage that you could clip those alligators directly to the pins. Perhaps there is a break in the wires from the pickups/wheel trucks to those pins.
