Sure. But this is a huge, incredibly novel prediction. It's not some trivial matter.
Once you grant that the metric varies smoothly over space and time, gravitational waves are pretty mundane.
Einstein for many years went back and forth on the issue of whether gravitational plane waves could be observed even in principle (sometimes accepting his initial findings, sometimes accepting Eddington's point that gravitational plane waves were unphysical artifacts of the approximation to the full theory Einstein had to use for calculational tractability, sometimes arguing that plane waves were physical but unstable, and several variations on all of these).
It wasn't until shortly after his death that it became widely accepted that gravitational radiation from a binary could couple to mass at a distance (notably Pirani 1956, reprinted <https://sci-hub.se/https://link.springer.com/article/10.1007...>, the two paragraphs after eq (2.17) on the 9th page of the linked pdf, p. 1223[1], and the three paragraphs after eq (3.7) on the 12th&13th pdf page, pp 1226-1227 being particularly interesting), which in turn provoked Feynman's sticky bead argument and the work by Bondi, Synge and others.
I don't understand the point your comment's parent's user has been trying to make in this conversation, but the entire enterprise of Einstein's theory of gravitation did not in its early days depend sensitively on the details (or even physicality) of gravitational radiation. Indeed, it was not widely accepted during his lifetime that gravitational waves could be measured even in principle. And even then it was almost twenty years before the first relativistic binary (Hulse-Taylor) was discovered and a further six or seven before it became clear that gravitational radiation is a useful test of various theories of gravitation. <https://ui.adsabs.harvard.edu/abs/1982ApJ...253..908T/abstra...> (PDF freely available via link in top left): "With the exception of general relativity and the Brans-Dicke theory, none of [these alternative gravitational] theories predicts even the proper sign of orbital period change due to emission of gravitational radiation, let alone the proper magnitude".
I would insert the words "local, minimal" before your "coupling between the metric and the stress-energy tensor". c, whatever its value, is relevant, and there are plenty of gravity theories with non-minimal couplings (NMDC) and spooky action at a distance (https://www.phy.olemiss.edu/~luca/Topics/ft/nonlocal.html is a "zoo"), which tend to have different post-Newtonian effects. I would almost add "unique" as a qualifier too, but e.g. Brans-Dicke is still technically viable.
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[1] Of particular (pardon the pun) interest to anyone who has read Synge's five point curvature detector.