I'd say the whole point of GAN is that generation is cheaper than classification, therefore an effective brute-force way of making a good classifier is to generate an infinite supply of examples with a-priori known classification, and pit it against a classifier.

Yeah, but the theoretical endpoint of training a GAN is that the generator gets so good that the discriminator has to resort to guesses and become unable to tell with any sort of accuracy as to whether the example it is shown is real or generated.