Well we are looking for a voltage drop, when you take the box out of the line check the voltage at the light, also see if you leave that connector unplugged what happens, do the rear light not work at all?
The left turn signal wire is violet, see if there is a violet wire in the connector, unplug it, and check for voltage on the female side, it should be 12 v.
I see a violet wire going to that wire nut, check that see if it is 12v when the left turn is on, off when the turn is off.
You say you cant find where that wire goes?
Even with leds in the rear the turn signal should come on and not necessary flash. It will start as normal then start flashing real fast to let you know it has a bulb out.
Message me your e-mail I will send you a schematic.
Or my e-mail is on the bottom of my signature.