Success!
My theory was correct! I dug into my dash this weekend and removed the
two pods that control the air scoop, and they were both shot!
I took them out and plugged the vacuum lines going to them, and my center vent has worked properly ever since!
Since my car has no passenger airbag, getting to them was very easy. All I had to do was pull out the glovebox, and the whole thing took less than 10 minutes to do.
I decided to pull apart the vacuum pods, and the results of that can be seen here:
http://www.youtube.com/watch?v=sovTANrf3pM
So,
If you have a W126 that has this strange center vent issue, I strongly suggest you start here first, since these are the easiest pods to get to, and are also the most likely culprit.
Cheers!