> we already know that GR isn't the whole story when it comes to BHs. See "soft hair" black holes
More precisely, most physicists believe that GR isn't the whole story. But we have no actual evidence for quantum gravity speculations like "soft hair". They're just speculations at this point. We don't know that any of them will actually turn out to be right.
> he first assertion, that GR is not the whole story, is more or less a given knowing that GR returns non-physical infinities when trying to describe what's inside the BH.
Only at the singularity, but the singularity itself is not even part of the spacetime manifold in GR.
In cases like black holes, there are physical invariants that do increase without bound as the singularity is approached, i.e., still within the spacetime, but their values are still finite at every point within the spacetime.
It is true that most physicists believe that when those invariants reach some particular scale, such as the Planck scale, the GR description in terms of spacetime geometry will break down. But that scale is about twenty orders of magnitude away from what we can currently probe observationally, so this is another of those speculations that, however plausible it seems, is not going to be testable any time soon.
You are right. GR being or not the whole story is a thing, "soft hair" black holes is another thing with a much more speculative edge.
But I would say that the first assertion, that GR is not the whole story, is more or less a given knowing that GR returns non-physical infinities when trying to describe what's inside the BH.