The tl;dr is that often one wants to know about an event
on the past or future light cone of an initial event, like connecting the detection of a soft X-ray with an inverse Compton scattering at some astrophysical source like a black hole's corona or high-energy galaxy cluster electrons (the Sunyaev-Zel'dovich effect), where one wants to trace out the evolution of the scattered photon's momentum across cosmological distances. Coordinate systems along the photon's path from the thermal emission source (the black hole accretion disc or the surface of last scattering) can facilitate this. These coordinate systems are reference frames moving at c.
As long as one is working with covariant descriptions of matter, taking a notion like a photon's "affine time" or similar for a classical electromagnetic wave into (signed) momentum imparted to some object on a timelike path (such an object will have rest mass) is straightforward. The justifications mainly originate with Newman & Penrose in the early 1960s.
Here's a nice (and very fresh -- it's an incomplete draft) technical note that details one use of the Newman-Penrose formalism in the flat spacetime of Special Relativity. Note the extract from Chandrasekhar's 1983 textbook in section 7.3 (the author sources from a 1998 reprinted edition). https://astromontgeron.fr/SR-Penrose.pdf> (PDF). The observations in §7.7 and §7.9 would be enough for me, if I hadn't already internalized the ideas further below.
I'll raise a couple of important and noncontroversial points from Jacques Fric's note: a photon -- whose frequency is proportional to its momentum -- oscillates some number of times along its null path lending a useful affine parametrization of the geodesic (photon motion is for all practical purposes always geodesic) that takes into account the spacetime curvature along the geodesic. Deploying the NP formalism in such a situation gives a nice analogy between motion through curved spacetime and motion through a refractive medium. And finally, covariant results in the NP coordinate system are readily interconvertible with covariant results in Minkowski coordinates.
Coordinate systems built along null geodesics -- typically lightlike Fermi-normal coordinates (FNC) -- also find applications in generalizations of the Jacobi equation, geodesic deviation, conjugate points, and so forth.
Now a step back to your comment. Null-basis FNC approaches let one calculate the momentum of a massless force carrier at any point along its evolution from p_{emited} to p_{observed}, and find cosmological applications (redshift of an astrophysical megamaser, the Lyman-alpha forest, etc) and microscopic ones (for the latter see ref [1] at https://physics.stackexchange.com/questions/62488/local-iner...>).
In reading that stack exchange answer you'll want to know that 'Penrose showed that any [Einsteinian] spacetime [...] has a limit which is a plane wave, which can be thought of as a "first order approximation" to the spacetime along a null geodesic.' (from https://link.springer.com/referenceworkentry/10.1007/1-4020-...>).
Pointing to references rooted in string theory and supersymmetry are not endorsements of those families of microscopic theories; they're just the most accessible examples of the practical use of lightlike FNCs for small locally relativistic systems.
So, not a logical absurdity, not useless, and not even especially uncommon.