You are right.
> It seems obvious to me that Navier-Stokes can't be a perfectly accurate description of reality, so it should come as no surprise if they turn out to blow up or otherwise behave non-physically.
The mapping between physical reality and Navier-Stokes equations is extremely well understood. We know when they represent an excellent description of the physical world, and we know when they could fail due to the finite number of particle in the fluid.
We also know how they fail, and can estimate corrections due to finite particle number, see for instance the Cunningham correction factor.
We also know that a much more complicated theory, Boltzmann transport equations, would be exact.
It would be surprising if Navier-Stokes equations blew up, because we know they represent most often physical reality up to extremely small, controlled corrections.
> Am I missing something? Why isn't it assumed that they're just a very nice approximation, like Newton's laws pre-relativity? Actually the whole thing reminds me of the ultraviolet catastrophe that preceded quantum physics.
Not really: the ultraviolet catastrophe was a symptom of unknown physics awaiting to be discovered.
We know very well the physics beyond Navier-Stokes, and still we use Navier-Stokes equations for their incredible effectiveness.
Using the full theory (Boltzmann transport equations) would make even the simplest fluidodinamics calculation virtually impossible, while adding a correction on the n-th significant digit, with n much higher than the precision one could reasonably expect.