General relativity predicts gravitational waves. For the theory to be correct, gravitational waves need to exist. You haven't fully confirmed the theory experimentally unless you confirm that consequence.
I don't see any possible basis for that belief (at the time) other than arguments based on non-empirical reasons like consistency and parsimony. (That is, it would be strange if other predictions of the model worked out but this one didn't.) Yet it is exactly this kind of theoretical reasoning that Hossenfelder, taken literally, seems to reject (in the absence of experimental data).
I'll also note that any other solution of the black hole information paradox has to be consistent with the the rest of what we know about physics, and any future empirical observations. So it can't be completely untethered from reality; it makes testable claims in this way. Further, it's not clear such solutions can never be directly tested, or will never be found to imply novel testable consequences.
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.