My dog loves pillows. But he doesn't move them. If he's laying down, and a pillow-like object is near, he'll use it. But if the sun moves, he'll move, without the pillow.
Humans are environment-changers. That’s why dogs teamed up with humans. Humans seem to spend a lot of time doing strange and seemingly useless things, like banging rocks together and looking at shiny boxes, but at the end of the day there is always extra food and the environment around humans is full of mysteriously comfortable objects.
If our dog wants to rest and a fleece or wool blanket happens to be in reach, it will pull it in position (and sometimes even fold it) in order to rest its snout on it. But admittedly it does not move the blanket substantially. I have yet to understand how it decides where to rest though. There is a lot of places and none seems to really dominate the other ones.
I have been paying more attention and ours loves head rests. So maybe they don't understand how to make or move them, he definitely prefers them. Disputing the original claim, or dogs differ too much from primates.