I would argue that GAN's by definition aren't unsupervised, they just aren't supervised by humans. Additionally, OpenAI's game stuff also has similar arguments against it.
I'm not sure that's correct. The discriminator and the generator both learn to match a training set. You don't need to label the training set at all. You can just throw 70,000 aligned photos at it.
I think I see what you're saying, but that might be a different definition of "supervised". It seems impossible for one half of the same algorithm to be supervised and the other to be unsupervised. But I like your definition (if it was renamed to something else) because you're right that the discriminator is the only thing that pays attention to the training data, whereas the generator does not.
You can define the terms how you want - but in terms of how they're understood in both industry and academia, you are incorrect.