> To my understanding, as a dilettante in the science of fluids, turbulence is an open problem mostly only in subfields pretty far removed from everyday engineering:
> The other group I’m aware of are those who study large-scale turbulence. The goal there is to summarize or coarse-grain the turbulence that goes on within a small computational volume, as a cost-saving measure versus using a more finely-resolved mesh in space and time. [...] But... it’s kinda phenomenological. Not as much fun for many scientists.
The turbulence "closure" problem you describe here is the main obstacle for the prediction of any turbulent flow. This is not far removed from everyday engineering! Practically speaking, the computational complexity of turbulence is far too high, so approximations become necessary.
You've described the basic idea behind large-eddy simulation, but I think you underestimate how interesting the theory could be there. Look into spectral theories of turbulence. I find this sort of research rather interesting, though it's formidable. My understanding is that some of the techniques Kraichnan applied to turbulence in the late 1950s were later independently redeveloped by quantum field theorists in the 1970s. (Note that I'm no expert in these models, but learning about them is on my TODO list.)
> The mathematicians question whether the Navier-Stokes continuum fluid model can be trusted to remain mathematically well-posed in all scenarios
The media unfortunately gives the wrong impression about the Navier-Stokes existence and uniqueness problem. This problem doesn't have much to do with turbulence in the computational complexity sense. I honestly don't see how proving that the Navier-Stokes equations do or do not have unique solutions is going to help turbulence. They've already proved that for 2D turbulence, but that didn't help solve 2D turbulence.
Anyway, your understanding of the problem is basically correct if not focused on the best examples. It's already known that the solutions aren't unique in inviscid compressible flows, but that doesn't stop people from using the compressible Euler equations as a model. They just add an extra condition to make the solutions unique (arguing that the other solutions won't appear in reality). My impression is that many people working on the NS existence problem believe that in certain circumstances the dissipation can become unbounded, and would be limited by different physical mechanisms that should be used instead. Practically speaking this might solved by simply using a different viscosity model, for instance.