Why do mathematicans care so much about the incompressible Navier-Stokes equations?

Two different answers:

First, it is an interesting mathematical problem unto itself. The mathematics community cares a lot about problems that are really not applicable to the real world.

Second, in applied mathematics we spend a lot of time looking at extreme limits. For example, in statistical mechanics, we frequently consider the "thermodynamic limit", wherein we work with an arbitrarily large number of particles and appropriately scale the other variables (for instance volume) to get the right behavior when we take the limit of infinitely many particles. In reality there are no systems with infinitely many particles, but at the same time this approximation is extremely accurate because our ordinary macroscopic objects are comprised of so many small components.

Using Navier-Stokes in the first place is very similar to this thermodynamic limit: real fluids are made of atoms, not continuum material, but there are so many atoms that thinking of them as continuous is usually accurate and makes the problem much more tractable.

The use of the incompressibility approximation in the Navier-Stokes equations is similar: real fluids are all compressible, but the compressibility of many important fluids, like water under typical conditions, is so astonishingly small that the incompressible approximation is reasonable, provided the external force is not too huge and neither the compressible solution nor the incompressible solution has any singularities.

To someone who likes pure mathematics, in a sense that he does not need that the solutions of certain mathematical problems have practical applications in other fields, a problem like Navier-Stokes problem could be interesting in itself as some kind of gem inside the field of partial differential equations.

Although some problems inside mathematics were formulated with the motivation that was somehow based on the practical aspects they still can be viewed as problems which reside inside certain mathematical field, and they are worth something even if they do not have practical applications at all.

This maybe could have been a few comments but as I think that is better that as many questions as possible have an answer I wrote this as an answer.

Edit after your update: Is it realistic to expect that if the incompressible case is not solved that compressible case could be solved before if compressible case looks more complex than incompressible? Although both cases are approximations it is somehow natural to try to prove first the easier case and then go into more generality, but if you think that there is some way that could first settle the case that looks harder and more complex, in which density is not constant, then I wish you luck.