> From the outside, though, is it even in principle possible to tell whether a BH is a stellar-collapse Bh or if it's an eternal Schwarschild vacuum BH?
There's a hint in "vacuum". There's real stress-energy and it's not where you would put it if building an exact BH solution by hand (I'll return to that in the last paragraph).
How do you tell if a body under a sheet has died peacefully in bed or was violently axe-murdered without lifting the sheet? Look for blood spatter.
If you see the daughter products of a failed core collapse supernova around a black hole, I think it would be strange to think, "hm, that black hole was probably there before the hot dense phase of the universe". The idea of a cosmic supervillain mischievously arranging nebulae around eternal black holes is amusing.
Isolated black holes are trickier, especially as masses go up. How does one distinguish between primordial black holes from early overdensities in the whatever was around at GUT scales or higher vs ones passes through the throats in in a cosmology like the Caroll-Chen model? Unless we catch them evaporating or until we spot them forming, I don't know. Spotting primordial formations is not hopeless, they can't all form with exact spherical symmetry or with the to-be-balded lumps and us on unfavourable alignments, can they? There's bound to be some larger (near-)extremal eating a smaller BH somewhere in our past lightcone. So even if they're very early we should see impressions of the extremal-with-lump gravitational radiation in the relic fields.
> Do I recall properly from my GR class that once stellar collapse begins the outer shell reaches the singularity in a finite time?
Yes, details in MTW section 32. Finite and fast by human wristwatch proper time.
> I think it does come down to philosophy (if you're staying within GR) or some theory that resolves the singularity to say whether the BH is "made of matter" or not
I think BH specialists would love it (and hate it) if someone found something in the matter sector that manifests truly enormous degeneracy pressure. Who knows what the heck is in the inner layers of neutron stars. Cutaway diagrams that show anything other than a ? near the core are wild speculations. One I saw that I enjoyed had six ?????? starting around 10^15 g cm^-3 just for emphasis. Unfortunately this wild hope gets ridiculously wild when considering the most massive known galaxy centre BHs, and eventually your explosion of question marks practically demand some quantum gravity (or asymptotic safety or something).
And anyway there are IR problems in quantum fields on general spacetimes. Even in extremely flat space, G = easily blows up, and who knows what we'll see as we develop devices to point to the source of weak gravity. How small a mass can avoid being in an eigenstate of position for a brief test? [arXiv:1602.07539 is just the start of that story!]
And furthermore actually solving the EFEs is a pain and numerical methods are still barely an aspirin, and anyway readily leads one into even more ways to mislead yourself if you don't cling to a T-first approach instead of a g-first approach ('t Hooft put out a pretty crazy seeming argument based on a brute force diagonalization recently). Sure one could argue that "matter determines curvature" and not the reverse is at least partly a philosophical point, but practically, even if you start with a ridiculously improbable stress-energy distribution you won't be chasing down regions of spacetime in which the eigenvalues of T_ij have the wrong sign. It is perversely common that when one writes down a metric first and then add matter, you end up with a proliferation of negative energy density or find lots of tension around extended objects, or the like.
Tl;dr: I look forward to a successor to GR, but am pretty sure that whatever it is will be even harder to teach.