Some thoughts: expansion-in-series-based methods (including Hilbert's, which is not used in practice) and the Chapman-Enskog method work only for moderately rarefied gas flows (where you can neglect higher-order collisions; this can be derived explicitly using the BBGKY hierarchy). Also, since the Chapman-Enskog method is asymptotic, it is not guaranteed that higher-order equations (inviscid Euler equations being the zero-order equations and Navier-Stokes equations being the first-order equations) will provide an accurate description of flows. Indeed, the second-order equations (Burnett and super-Burnett equations) seem to fail in some cases, while providing more correct results in others. But given the complexity of the equations themselves and the complexity of the boundary conditions, no one really uses them. The cool thing about the Chapman-Enskog method is that it gives a closed set of equations, so you don't need empirical models for heat conductivity, viscosity, etc.

That's the first point – that methods depending on series decomposition might never guarantee a solution that's accurate in all cases. There are also moment-based methods (Grad's method, for example, being one of the most famous), which have additional equations for parts of the stress tensor (I think; never really read much about them).
The second point is that the equations correspond to conservation laws: mass, linear momentum, energy. The equation corresponding to the conservation of angular momentum is usually neglected: the terms related to internal angular momenta of particles are considered to cancel each other out (which seems logical, since unless there's some magnetization happening, the particles will be chaotically oriented and the average of the angular momentum will be 0), and in that case, the equation is satisfied since it just follows from the equation corresponding to the conservation of linear momentum.
However, there's been some research recently on whether this equation can actually be neglected and what implications it carries, whether it's connected to turbulence or some other effects.

The third point is that in high-altitude hypersonic flows, there are far more complex effects going on in flows that just simple collisions between particles – there are transitions of internal energy (which is a quantity described by quantum mechanics), chemical reactions (dissociation, exchange reactions), and this all complicates the Navier-Stokes equations – additional terms appear (bulk viscosity, relaxation terms, relaxation pressure). And correct modelling of these terms requires solving large linear systems with quite complex coefficients, and to complicate things further, for many of the processes mentioned, there aren't any easy or even correct models (to take into account dissociation, for example, you need to know the cross-section of the reaction for each vibrational level of each molecular species involved in the flow), since these models are either computed via quantum mechanics (which takes enormous amounts of computational power) or are obtained experimentally (which limits the range of conditions under which the results are obtained).

DSMC methods have being increasingly popular as of late, but of course, they can't provide theoretical results, while it is possible to observe some interesting effects even in theory using the Chapman-Enskog method.

So the problem is not only getting more "correct" equations, it's also being able to correctly model everything that goes into the equations we currently have, and then being able to solve them (for a simple flow of a N2/N mixture, if you use a detailed description of the flow, you get a system of 51 PDEs). And in engineering applications drastically over-simplified models are often used, and yet it's not like every high-altitude air/space-craft has burned to a crisp because of this. While new, "more correct" equations are interesting, of course, there's enough work to be done with the current ones.

Source: I do theoretical research and numeric computations of rarefied gas flows for a living (at the Saint-Petersburg State University).