That wire has been repaired in the past. This may seem silly, but run a jumper wire from the light housing to another good ground on the body/frame of the car. Does light work now? Check light bulb- it should have two filaments in it with two contacts on the bottom. It should have two offset locating pins. When installed, do the contacts on the bulb line up with the contacts in the socket? Lets check basics before we move on. Good luck.
So I just tried that and i am still getting the same result. Also I forgot to mention that when the headlight switch is on and the right turn signal is on the rear passenger turn signal also does not blink. It comes on but does not blink.
And also here are pictures of the contacts of both front signals.
Passenger side
Driver side