Well, even though this is resurrected from 2016 it seems to have captured my attention for a bit. :-) Before this HN posting I had not read the article nor the paper it discusses. (I had read all the other linked items below.)
The arxiv version of the paper <https://arxiv.org/abs/1604.05507> (Uggerhøj et al 2016) without saying so in words does post-Newtonian ("PN") corrections to Poisson's equation for Newtonian gravitation <https://en.wikipedia.org/wiki/Poisson%27s_equation#Newtonian...> (note the last equation in that section is the same as the paper's eqn (1)). The corrections enter with constant multiplications by c^2 and the reciprocal. Wikipedia is not helpful: <https://en.wikipedia.org/wiki/Post-Newtonian_expansion#Expan...> is only a paragraph long. Clifford Will's 2011 PN paper is an excellent overview, and the first two pages or so (of nine total) are reasonably accessible to undergrads and laypersons (arxiv version: <https://arxiv.org/abs/1102.5192> and from there you can get a link to the published PNAS version if you somehow prefer that interface). For laypersons Gerhard Schäfer's "Step by step from newton to Einstein" <https://www.einstein-online.info/en/spotlight/postnewton/> is very good.
There is nothing wrong with the authors (or Feynman) treating the relativistic age difference as a problem approximately solved using PN methods. (I'll justify that near the end of this comment). Probably that's what Feynman did, and of course Einstein essentially invented PN methods due to a lack of computers of any sort to help solve problems in the full theory of General Relativity. The Uggerhøj et al. 2016 paper itself somewhat too carefully restricts itself to expressions like "time dilation in relativity" and "relativistic aging", never outright deploying the words "General Relativity" or its exact synonyms, without explaining why. The abstract goes a bit too far with this, imho, "... a fairly complete solution to the relativity of 'aging' of an object due to differences in the gravitational potential" is strictly true -- although "fairly" does some heavy lifting -- and "differences in gravitational potential" is simply not from General Relativity and its curved spacetime, but rather from Poisson's equation for Newtonian gravity with corrections for the finite speed of the (Newtonian) force of gravity.
I think Uggerhøj et al could reasonably expect a reader to misunderstand their paper as an analysis using General Relativity, and indeed that's exactly what Jacob Aron (the author of the New Scientist article) did: "Physicists have calculated ... the effects of gravity as described by general relativity." No. They have not.
(Does this matter? Yes, the moment you introduce an observer moving faster than a few percent of c, you get can get bad results from expansions in orders of v^2/c^2. Does this matter for the purposes of using depth-dependent differences in isotope abundances as a notion of "age", which was the focus of Uggerhøj et al. and (according to them) Feynman? No, probably not. But if you had a Earth-observing neutrino satellite sensitive to beta decays at depths from the surface to the core, and that satellite were in low Earth orbit, you'd want a check on any PN methods relying on v being much less than c (again, see near the end of this comment) which might be used in analysing the observed beta decay spectrum. Not total sci-fi: for example Borexino <https://borex.lngs.infn.it/> picks up (anti-)neutrinos from the Earth, and smaller detectors of "geoneutrinos" is an interesting area of research).
The central message of Uggerhøj et al paper (which the New Scientist article gets acceptably right) is that sources are important in gravitational physics, and that a better description of the Earth's bulk composition gives more reliable results. That's true in the PN case (as discussed in the article) as well as well as what we'd want to do in General Relativity: set out the stress-energy tensor (a description of the matter and its non-gravitational interactions) and feeding that into the Einstein Field Equations to extract the metric tensor. Even today we can only attempt the latter numerically with supercomputers; there was no realistic path to such a solution in the 1960s.
But for most practical purposes (including GPS and other GNSS) we are fine with low-order PN corrections for the Earth's gravitational field (measured empirically using artificial satellites, lunar laser ranging, and other tools in geodesy and geodynamics) -- see for example the excellent Soffel & Frutos 2016 "On the usefulness of relativistic space-times for the description of the Earth's gravitational field", <https://link.springer.com/article/10.1007/s00190-016-0927-4> which (because Springer sucks) you should obtain via s c y h o b <https://sci-hub.se/https://link.springer.com/article/10.1007...>, which makes the case that "the first post-Newtonian approximation is completely sufficient for a description of the Earth's gravitational field and also for a description of satellite orbits" and "the use of a higher post-Newtonian or even exact metric [i.e., General Relativity] is not justified as long as a theory of relativistic systems is not available at [present day] level of accuracy."
Finally, anyone comfortable with gravitational theory is likely to enjoy Philipp et al. 2020 https://doi.org/10.1103/PhysRevD.101.064032 aka https://arxiv.org/abs/1912.10159 ("Comparing the Earth’s Newtonian geoid to its relativistic generalization is a very subtle problem, but of high interest")