How to handle this? Well, as Andy over at Patience Corners noted,
I am told that mountain climbers will occasionally put themselves into positions where the only way they can go is up. "When the only way you can go is up," I read, "you jolly well go up."All the wires are coming out and I'm redoing the connectors. There will be no turning back now.
Update: Glorious success! I'm replacing all connections with either molex pin connectors or spade connectors and I just got my tail lights to light up using a 12V power supply and the new connectors. I know it's like proving that gravity still exists, but it's heartening nonetheless.