Maybe a working quantum algorithm for weather prediction would outperform currently used classical simulations, but I wouldn't expect it to be bang on every time. Inputs are imperfect. So at best you could benchmark it, and gain some confidence over time. It could very well be good enough for weather prediction though.

Also I doubt that a quantum algorithm is possible that provably solves the Navier-Stokes equations with known boundary and initial conditions. At least you need some discretization, and maybe you can get a quantum algorithm that provably converges to the real solution (which alone would be a breakthrough, I believe). Then you need some experimental lab setup with well controlled boundary and initial conditions that you can measure against.

In any case the validation would be at a very different standard compared to verifying prime factorization. At most you can gain confidence in the correctness of the simulation, but never absolute certainty.

At scale, yes. But this would still be solving toy problems with less variables, fewer dimensions.

And they’re not actually solving weather problems right now, I think. That was just an example. What they are actually solving are toy mathematical challenges.