The only one I can think of is simulating physical systems, especially quantum ones.

Google's supremacy claim didn't impress me; besides being a computationally uninteresting problem, it really just motivated the supercomputer people to improve their algorithms.

To really establish this field as a viable going concern probably needs somebody to do "something" with quantum that is experimentally verifiable but not computable classically, and is a useful computation.

That is equivalent to proving BQP ≠ P. We currently don't know that any problem even exists that can be solved efficiently (in polynomial time) by quantum computers but not by classical computers.