In re-reading your post, codi, maybe I can suggest another thought.
You see a black wire melted at the plug. There are, at least, five filaments, two tails, two stops, and a reverse. With brake power applied, (tails are out), stops are on, rev gear is selected, you have three heavyweight bulbs lit, a fair amount of current flowing.
There is a single wire supplying each filament, but, a single wire for return for ALL three bulbs. Hence, the current flow is alittle heavy for this one wire which is the same AWG as supply. The connector pin for this ground gets warm, expands, causes corrosion, now becomes a resistor, current flows generates more heat, eventually would burn up and open.
It will be interesting to read what the problem is/was after others have looked and investigated the challenge.