It is worth recognizing that the argument being made is something of a "god of the gaps" argument. In other words, our models of neutron stars aren't terribly precise (e.g. predictions of the maximum neutron star mass based upon QCD a.k.a. quantum chromodynamics which is the Standard Model theory of the strong force, and nuclear physics, and general relativity (GR) have relative uncertainties on the order of 10-20%).
So, any "new physics" that would impact the structure and evolution of neutron stars that has an effect smaller than this uncertainty can't be ruled out, even though there is no positive evidence that axions are present in neutron stars or even exist. An argument that "new physics" exists because we can't rule it out, even though we have really no positive evidence for it, is a weak argument.
One of the
more clever ways proposed to indirectly identify the existence of dark matter, but not its exact type, is to look for systemic differences in the properties of neutron stars between the vicinity of our solar system near the rim of the Milky Way and the galactic center where the proportion of dark matter in neutron stars should be greater.
Efforts to estimate optimal parameters for "axion-like particles" that are ultralight typically come up with values for their mass more than ten orders of magnitude smaller than the mass that a QCD-axion is expected to have. (See, e.g.,
here at Section 3.5)
QCD-axions have some features that make them desirable dark matter candidates, nonetheless, because particles much below a keV mass can't be created via thermal freeze-out in the early universe, because otherwise they would have too high a mean velocity and would constitute "hot dark matter" which is inconsistent with the amount of large scale structure observed in the universe. Instead, lighter particles that constitute dark matter must be created and destroyed in equilibrium amounts over the course of the universe's life with fairly low (non-relativistic) momentum. QCD-axions are a candidate that would clearly fit that requirement.
On the other hand, numerous experiments aimed at directly detecting QCD-axions have come up empty. There has been
lots of scholarly work on this hypothesis, but nothing so far to give it a decisive advantage over other possibilities.