Ok so I have a question, does the Passat act like an old American car? When the turn signal bulb is burned out the blinker won't work.
I personally don't know if this is the case on the Passat as I have never had an issue with the turn signals.

 

Modern cars will blink twice as fast when a bulb is blown. The bulbs light up when I turn on the hazards, so it's not the bulbs.

 

Based on your description and everything you've changed, the only possibility I see remaining is the wire from the turn signal switch to the splice in the instrument panel wiring harness. It could be a bad connection at the turn signal switch connector, the splice in the wiring harness, or maybe the wire is damaged along the way, but it can't be more that a few feet long!

The wire in question is black/green, position 11 on the turn signal switch connector.
It also appears at the left A-pillar connector station, beside the driver's left foot, on the 10-pin brown connector, position 10 (also black/green). (Third row, near the left side, connector T10a.)
I would check for continuity between the turn signal switch connector and the A-pillar connector. If you don't have it, run a wire between those two points.

The splice in the instrument panel harness is almost certainly very buried and in an awful place to access.
There are other places besides the left A-pillar, but it's the closest to the turn signal switch.

The wire is 1.0mm^2, about 16 AWG. Be sure to use fine-stranded wire for the repair (not building wire).
Sounds like a job for a 3M tap-and-run...

References:
Bentley schematic 48/10-16, p. W97-3

 

Thanks. I will check at that connector. Definitely didn't want to dig through the harness with all that wrap around it.

 

Dug around the car some more and checked for continuity from the turn signal plug and the brown plug at the bottom of the A-pillar. No continuity. Tried to find a break somewhere along the line but wasn't able to see anything; even after moving the relay box. Ended up just running a new wire from the turn signal plug down to the brown plug and I now have working blinkers again. Thanks again.

 

Now how does something like that just happen? It's a rhetorical question. Glad you got it working again.

 

I think that splice is under the front carpet (LHS).

 

You've got a blown fuse. Same thing happened to mine. Quick fix!

 

No, a blown fuse won't disable turn signals on only one side. Read the resolution above.

 

I thought maybe he had declined when the dealership told him he needed a blinker fluid change.

 

My dad told me about a guy in the Air Force that was so gullible, that they'd send him to the shop for a can of striped-paint, or a left-handed smoke changer.