Bob, from your reading, do you know if this white residue is supposed to be a plastizer that is spreading to the plastic surface?
That could be a bummer to remove without damaging the surface.
That's my understanding of what it is (plasticizer) and yeah, it can be a pain to remove -- but it's not impossible. I've run across this in a few vehicles including an Acadia, a C5 Corvette, and perhaps worst of all a first-gen CTS dashboard (those familiar with the original CTS know why the dashboard is a nightmare to clean/detail).
It's not impossible to take care of safely, but it's also labor intensive to scrub everything down. My method is to mix up a batch of cleaning solution with a little Woolite detergent, some APC (I usually use Simple Green, but any APC should work), some OxiClean powder, and some extremely hot water.
Scrub all the dash/door panels/other vinyl, rubber, and leather surfaces with an MF towel that has been soaked in the bucket of cleaning solution, using a soft tooth brush to scrub any stubborn areas further. I then follow up with my vapor steamer, removing runoff and drying the panel with either another MF or a cotton terry towel.
The white substance is definitely a sprayed on product. I can see run lines from the hard plastic part of the door panel to the leather insert on the door panel. I was able to remove a little on an inconspicuous area using "Goo Gone". I just don't want to use it all over if it could damage the finish.
What I would suggest is that if Goo Gone is working for you, go ahead and use it to remove the residue, but IMMEDIATELY rinse it with another cleaner and apply protectant -- I'd say with about 99% certainty that you aren't going to permanently damage the door panel with the goo gone, but I can't guarantee whatever the stuff you're removing is hasn't stained or etched the door already.
You could also attempt using the cleaning solution I described above, substituting a spray-down with plain hot water to rinse if you don't have a vapor steamer at your disposal.