My guess is you had an intermittent-partial "weak" short-circuit in the wiring. By "weak" I mean perhaps just one tiny strand of a stranded wire that is touching some other contact or terminal...or something of that ilk. It wasn't a "strong" short-circuit otherwise your breaker would have popped. The weak short-circuit might be lowering the voltage to the ITAD and any nearby signals...perhaps explaining why the green light is on but dim...and if the voltage to the ITAD is too low I can imagine a scenario where it causes the relay to behave intermittently (chatter).
Early on you said physically rotating the ITAD changed the behavior. I'm thinking that movement pulled-on or jostled the cable/wires "just enough" to break the weak short-circuit. Likewise, removing the billboard wire was also such a movement.
Clearly I'm wildly speculating. I was trying to come up with a plausible explanation for all the behaviors you reported. I suppose we could play a game of 20 questions asking you to try this experiment or that experiment. But if it's now working not sure there's much point.
Presumably you found a different point to connect the billboard and it is working again too.