Okay, after figuring out which paper you were trying to link to in the third link, I decided that it is better to just give an answer rather then a bunch of comments. So... there are several issues at large in your question. I hope I can address at least some of them.
The "big picture" problem you are implicitly getting at is the Hilbert problem of hydrodynamical limit of the Boltzmann equations: that intuitively the ensemble behaviour at the large, as model by a fluid as a vector field on a continuum, should be derivable from the individual behaviour of particles, as described by kinetic theory. Very loosely tied to this is the problem of global existence and regularity of Navier-Stokes.
If your goal is to solve the Navier-Stokes problem using the hydrodynamic limit, then you need to show that (a) there are globally unique classical solutions to the the Boltzmann equations and (b) that they converge in a suitably regular norm, in some rescaling limit, to a solution of Navier-Stokes. Neither step is anywhere close to being done.
As far as I know, there are no large data, globally unique, classical solutions to the Boltzmann equation. Period. If we drop some of the conditions, then yes: for small data (perturbation of Maxwellian), the recent work of Gressman and Strain (0912.0888) and Ukai et al (0912.1426) solve the problem for long-range interactions (so not all collision kernels are available). If you drop the criterion of global, there are quite a bit of old literature on local solutions, and if you drop the criterion of unique and classical, you have the DiPerna-Lions solutions (which also imposes an angular-cutoff condition that is not completely physical).
The work of Golse and Saint-Raymond that you linked to establishes the following: that the weak solution of DiPerna-Lions weakly converges to the well-known weak solutions of Leray for the Navier-Stokes problem. While this, in some sense, solve the problem of Hilbert, it is rather hopeless for a scheme trying to show global properties of Navier-Stokes: the class of Leray solutions are non-unique.
As I see it, to go down this route, you'd need to (i) prove an analogue of DiPerna-Lions, or to get around it completely differently, and arrive at global classical and unique solutions for Boltzmann. This is a difficult problem, but I was told that a lot of very good people are working on it. Then you'd need (ii) also to prove an analogue of Golse-Saint-Raymond in a stronger topology, or you can use Golse-Saint-Raymond to first obtain a weak-limit that is a Leray solution, and then show somehow that regularity is preserved under this limiting process. This second step is also rather formidable.
I hope this somewhat answers your question.
I moved this question to math.SE a month ago. This is indeed the problem I got from the research, though it may not very appropriate here.
@Willie Wong gave a very nice answer to the question. Instead of closing or deleting the question, I think it's worth putting the link here.
Best Answer
In regards to the question of the "consensus" or "correctness", I will only point out that Tristan Buckmaster has had a proven record of studying nonuniqueness problems for low-regularity solutions in incompressible fluids, and contributed significantly to the settling of Onsager's Conjecture on the nonuniqueness problem for incompressible Euler.
In regards to Navier-Stokes: weak solutions are called weak for a reason. To put it in simplest terms: the "solvability" of a PDE depends on what you accept as a valid solution.
(As a digression, this is not a problem unique to PDEs. Even in arithmetic if you work over $\mathbb{Q}$ the equation $x^2 = 2$ is not solvable, and if you work over $\mathbb{R}$ the equation $x^2 = -1$ is not solvable. Mathematics has a long history of "completing" the "space of admissible solutions" to solve previously unsolvable problems.)
There's an obvious trade off: if you enlarge the admissible solution space, you make it easier to solve an equation. But by making it easier to find a solution, you risk making it possible to find more than one solution.
(As an example, consider $x^3 = 3$. It is not solvable in $\mathbb{Q}$, it has a unique solution in $\mathbb{R}$, and it has three solutions in $\mathbb{C}$.)
In some sense you can think of existence and uniqueness as competing demands; a lot of PDE theory is built on figuring out how to restrict to a reasonable set of "admissible solutions" while guaranteeing both existence AND uniqueness.
In the context of Navier-Stokes, Leray (and Hopf) figured out a way to guarantee existence. People however have long suspected that their method does not guarantee uniqueness (in other words, that they are too generous when admitting something as a solution). Buckmaster and Vicol's work tries to carve away at this problem, by proving that for an even more generous notion of solution non-uniqueness can arise.
So no, we are absolutely nowhere near saying anything useful about physics or engineering; we are merely calibrating PDE theory.
As an aside, local existence and uniqueness for smooth solutions of NS hold. So a "similar result for smooth solutions" is in fact, impossible. This brings me back to the point of calibration:
The main question on Navier-Stokes existence and uniqueness can be reformulated as: does there exist a sense of weak solution which guarantees global, unique solutions for all initial data, or is there a dichotomy where a sense of weak solutions that guarantees global solutions for all initial data is always too weak to guarantee uniqueness, and any sense of solutions guaranteeing uniqueness of solutions is always too strong to guarantee global solutions.