Humans make many choices that effect the trained weights: curation and choice of datasets, training schedules and hyperparameters. If these choices are made with an eye towards the generated results, rather than mechanically based upon test scores, why doesn't that rise up to the minimal level required to get a copyright on the weights?

It might. As I say, it’s up for debate. A judge might look at the 1kB of hyperparameters versus the 1TB of training data, and the 10 person-years of human effort versus 100,000 GPU-years of computer effort, and conclude differently.

Does Google have copyright of their search index? Never tested, as far as I know.