Why? Is this important as a sanity check in the absence of any independent verifications?

I'm not an expert on this, so take this with a grain of salt. Chaotic PDEs are extremely sensitive to initial conditions. This essentially makes it so that any numerical solution will (quickly) diverge from the true solution over time. (Just due to floating point error, discretization error, etc.) This is why for a lot of turbulent navier-stokes stuff, people don't necessarily care about the specific phenomena that occur, but look at statistical properties.

I think one of the reasons it is important to preserve conservation laws is that, at the very least, you can be confident that your solution satisfies whatever physical laws your PDE relies on, even if it's almost certainly not the "actual" solution to the PDE. You actually can ensure that a numerical solver will approximately satisfy conservation laws. Then at the very least, even if your solution diverges from the "actual" PDEs solution, you can have some confidence that it's still a useful exploration of possible states. If conservation laws are not preserved AND your solution diverges from the "actual" PDE solution, then you probably cannot be confident about the model's utility.