This seems to be a case when the principles, the foundations of a theory like GR can not be refuted by supposed consequences of something, like GW that is said to be derived from the theory, without incurring in contradiction or falacy.
It is important to remark that a geodesic itself is inevitably a somewhat idealized concept, like most habitually used concepts in physics without loss of generality and validity of the empirical results obtained from them. Since the trajectory a body describes is also a set of points, necessarily a massive body, no matter its size is idealized to a massive point when speaking about its location in the trajectory it draws, a timelike geodesic path or trajectory is a set of points at which the Chrystophel symbols can be made to vanish, irrespective of the size or mass of the body that describes that path, that path is a one-dimensional curved line in curved spacetime. For reference see:
http://en.wikipedia.org/wiki/Normal_coordinates#Geodesic_normal_coordinates
This is notwithstanding the fact that in a body in motion for instance rotating there will be points that are obviously not describing the same path than the object to which they belong because they have motion relative to the center of mass of the object.
Here is a paragraph taken from the GR textbook of a relevant relativist that summarizes perfectly what I'm trying to get across
So going to the principles of GR one finds that:
"In Einstein’s General Relativity, gravity manifests itself by a tensor field and, in the absence of other forces, the motion of a particle is determined by this tensor field;
it does not depend on the mass of the particle. Einstein’s revolutionary idea is that the arena where the gravity tensor lives is itself determined by gravity, both are united in a 4-dimensional Lorentzian manifold (V, g), called the spacetime.
The trajectories of particles are geodesics of the metric g. Einstein’s weak equivalence principle corresponds to the fact that in a general Lorentzian spacetime the geodesic equations are formally the same as in Minkowski spacetime with arbitrary coordinates where the non-vanishing of the Christoffel symbols signals the presence of inertial forces due to the noninertialframe of reference.
In a general spacetime (V, g), it is always possible at one given point,
to choose local coordinates such that the Christoffel symbols vanish at that point; gravity and relative acceleration are then, at that point, exactly balanced. It is even possible to choose local coordinates such that the Christoffel symbols vanish along one given geodesic; astronauts in spacecraft have made popular the fact that in free fall one feels neither acceleration nor gravity;
in a small enough neighbourhood of a geodesic the relative accelerations of objects in free fall are approximately zero. Massive pointlike objects in free fall follow a timelike geodesic"
General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation.
From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.
The consequences this may have for GW are not for me to spell out with authority (I'd rather let everyone draw their own conclusions), since I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.
