My full-tail GSP had a great tail! It was a bit of a corkscrew 
The GQ answer to the OP’s question is yes.
Dogs have proprioception, knowledge of the relative position of all parts of their body.
Those you tube videos show cats carefully winding their way through pencils standing on end, and dogs knocking them over like crazy.
Perhaps they just don’t give a darn.
I’ll put the debate to rest. I asked my dog if she was always aware of when her tail was wagging and she responded “no”.
That sounds more like cats than dogs.
I have two border collies. One of them wags her tail all over the place, high in the air. The other keeps her tail tucked and only wags it low, but when she’s excited she wags so hard that her entire body moves. The other dog’s wags are mich more subdued.
Scottish sheep herders have a saying, “Foolish tail, foolish dog.” They claim that dogs that keep their tails down and in control are better sheepdogs. I have no idea if that’s true, but the saying has been around for decades.
The Nova Scotia Duck Tolling Retriever ‘tolls’ ducks by waving its tail around high in the air above the rushes.
I suspect wagging in dogs is something like smiling in people - something that happens somewhat automatically based in emotion, but is still controllable by the dog.