It's not really separate, because more things can fall in.
It's also not really separate if black holes may evaporate, in part because it is the outside conditions that determine when (and even if) evaporation begins. There are unsettled questions about whether black holes, even given suitable exterior conditions, fully evaporate; and exactly how to connect what fell into the black hole with what's left after complete evaporation. However, it is the mass inside that gives rise to Hawking radiation outside, so there is some connection.
Otherwise I think you're quite right, particularly about the gentleness of the Weyl curvature (the tidal part of the Riemann curvature tensor) around the event horizon of a large black hole.
"Pass through the event horizon, alive" is the "no-drama conjecture" that is part of the firewalls debate. Indeed, extremely massive black holes should have the least drama in the classical theory which describes black holes in the first place, so what feature of some quantum theory generates extremely high energy particles that we don't find anywhere between the Earth and Moon, or in the Jovian system, etc? "The firewall radiation can only be seen upon crossing an event horizon and there isn't one in those parts of our solar system" is not very satisfying, and it turns out that the arguably best-developed answer to that (by Afshordi/Dykaar/Abedi) is not significantly supported by gravitational wave data (so called repeating damped "echoes" cannot reliably be extracted from the noise) from LIGO, Virgo, and Kagra so far.
Though they might just be mathematical abstracts and not "real".
https://en.m.wikipedia.org/wiki/White_hole
https://physics.stackexchange.com/questions/740527/penrose-d...
What does that mean? The Schwarzschild radius is just the distance at which even light can no longer escape, the only matter there is the matter that happens to be falling in at that moment. Unless you mean the average density within the Schwarzschild volume (a term I just made up as far as I know, but you know what I mean) if all the mass of the black hole were spread out evenly throughout it.
That seems like a weird condition. "Average density" is pretty well-defined and doesn't need even spreading.
Even more, isn't the distribution of mass within a black hole a meaningless question? There's no information coming from the black hole, for an external observer it literally doesn't matter how the matter is arranged within.
A rotating (or merging) black hole can be deformed (non-spherical), but that doesn't stop you to calculate average density.
A longer answer is somewhat model-dependent, and somewhat on how one chooses to split spacetime to distinguish spatial distances (for volume) and time (the volume is time-dependent). Unfortunately this means thatg for practicaly any given black hole there is no unique definition of its interior volume.
In general it is fairly safe to say that within generic black holes there is a small volume that tends towards infinite density, and that small volume is embedded in a much larger (even for a tiny-mass black hole) interior space, even fairly early in the (time-dependent) black hole's lifetime (e.g. within a few horizon-diameter light-crossing times after formation by matter collapse).
The most commonly known theoretical models of black holes are, absent perturbation, not time-dependent. This tends to highlight the non-uniqueness of interior volume. DiNunno & Matzner 2008 <https://arxiv.org/abs/0801.1734> is a fine pedagogical treatment. 'An occasional question to the teacher of relativity is: ".. then what is the volume of a black hole?" The answer is that, unlike the response about the surface, the volume depends on the way that the 3-dimensional "constant-time" space containing the black hole is defined.'
I have a memory of a fine and surprisingly accessible treatment by a mathematician about the interior volume of a Schwarzschild BH, but unfortunately I can't find the URL. There are plenty of other discussions about BH interior volumes scattered around the web and the academic literature, although a depressing number of the latter focus on anti-de Sitter (AdS) black holes. Black holes in our universe, like the one in the linked Chandra article or like Phoenix A, are decidedly not embedded in a collapsing spacetime with a lightlike boundary "screen" as would be the case if the exterior spacetime of these black holes were AdS rather than expanding Friedmann-Lemaître-Robertson-Walker with local overdensities (the galaxies around these black holes). Which is too bad, because if our universe were AdS, the interior volume of a black hole could have a nifty relationship with information complexity; in our universe, shrug.
Two more things for completeness. The exterior of a black hole, if not vacuum, can be relevant in determining the interior volume. Single black hole exact solutions (Schwarzschild, Kerr-Newman) have infinite vacuum outside the horizon, but one can introduce perturbations (lumps of matter or other mass-energy) that raise "bumps" on the horizon and strangenesses in the interior. For a large and relativistic perturbation like a second black hole, or a swarm of them, you run into the problem that in general single black hole solutions do not superpose well (and certainly not linearly). The interiors of "hard" black hole binaries may be very different from that of black hole binaries so soft (or wide, thinking only spatially) that they are barely in mutual orbit. And finally, black holes may evaporate completely, so the spacetime would then be finite, and thus it would be weird to cut the interior spacetime up into time-indexed spatial volumes where some of those volumes are infinite.
Consequently, calculating an average density comparable to that of a homogeneous ball of fluid (or even a differentiated planet like Earth) doesn't really say much about the black hole itself. And that I think is the weird thing related to your comment.
Finally, even for round-planet-sized objects and stars, the interior volumes "suffer" a mainly mass-dependent volume surplus compared to a Euclidean 3-ball. The interior Schwarzschild solution is a starting point; it provides a calculation for the volume of a self-gravitating sphere of a constant-density fluid or gas and this calculation reveals that that volume is greater than the corresponding empty spherical shell <https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#...>.