> reduction in entropy

The formation of a black hole surface around ordinary matter is an increase in entropy if "no hair" is correct. All the individual masses and linear and angular momenta (and electric charge) are hidden behind the trapping surface. By examining that surface you can't tell how many bits and pieces there were inside when it formed initially or which fell in later; only the aggregate values are available. And all of those bits and pieces are crushed into a very small configuration (up to an outright singularity) compared to the volume of the interior. So the interior is mostly vacuum and vacuum is maximum entropy (details for the curious about how this works with a quantum rather than classical vacuum and black hole complementarity: <https://arxiv.org/abs/1310.7564v2>).

No-hair might not be correct though. Cf. Hawking's final interests in (<https://en.wikipedia.org/wiki/Bondi%E2%80%93Metzner%E2%80%93...>) superrotation and supertranslation "soft" (as in ~zero energy) hair.

> theories where black where black holes have a lot of entropy

Textbook black holes have a lot of entropy.

If "no hair" and a thermal Hawking spectrum up to final evaporation are both correct then low-entropy systems (like a chicken egg or a brain) become hopelessly scrambled and ultimately turned into greybody radiation from which one cannot even in principle determine the antecedent configurations.

That information loss is the upsetting thing, not the balding away of whatever bumps are raised on a black hole as the egg is thrown in balding away in ~ light-crossing time. The then more massive balded black hole (i.e., relaxed back into a "no hair" state) can be entirely represented by a tiny handful of numbers compared to the matter that formed it or fell into it later. Barring something like "soft hair" the numbers required to represent a star like our sun (and an egg) is much much higher than that of an egg thrown into a stellar mass black hole.

That's fine if the egg and all the rest of that stellar mass stays hidden inside the black hole forever. But with Hawking evaporation the shrinking black hole gives us no details of what was thrown in: ultimately, at final evaporation, we would have a stellar mass (and an egg mass) worth of almost entirely photons back. That wrecks unitarity, which is important to particle physicists.

The BMS-group supertranslation and superrotation idea is that the horizon wiggles a bit as the egg is thrown in, and that wiggle emits gravitational radiation with enough complexity in the waves to encode all the microscopic information in the egg (notably lepton number, baryon number, and strangeness).

> black holes have a lot of entropy

So does the infinite empty space surrounding them in a Schwarzschild or Kerr universe. Think like Boltzmann: take a volume of the totally empty space far from a black hole and swap it with a volume of totally empty space somewhere else outside the black hole. Does that make a non-negligible difference to the spacetime? Like, does it light something up, or does it change the geodesic equation? (A: No, not for these vacuum solutions. But it does make a difference if we swap some near-horizon relatively-Hawking-quanta filled space with some emptier far-from-horizon space (n.b., not a vacuum solution).)

Finally, the interior of vacuum-solution black holes (or Lemaître-Tolman-Bondi black holes formed by dust collapse, or other types of dynamical/evolving black holes surrounded by infinite vacuum) is a tiny volume compared to the exterior.

The most relevant volume outside black holes in our universe is the observable volume (set by the cosmic particle horizon), which will not be infinite at times in which there are black holes. There are many ~ billion-solar mass black holes inside the present ~ 10^{80} m^3 observable universe. In the future that volume will be larger, but so will the number of black holes in it, and almost certainly the masses of the largest black holes will be much bigger.

(In the even farther future, if total evaporation happens, there will be no black holes in the big-but-not-infinite observable volume centred on what was our galaxy cluster).