However, to an extremely good approximation, they'll still look like objects with just three properties.
See the table on this page for the categories:
Croker, Weiner et al. (the authors of the topic paper) are keen first and foremost on their spinning black hole interior solution, which is wholly classical and found at <https://arxiv.org/abs/2107.06643>. In the more recent topic paper they argue that they can make the exterior solution well-behaved too, following the path McVittie paved in 1933 in embedding massive objects in an expanding classical spacetime.
They don't come to the end of the path though. As they say in <https://iopscience.iop.org/article/10.3847/2041-8213/acb704> §4.6, their desired combination of interior solution, initial formation, infall/merger, arbitrary angular momentum, and being easy to embed in an expanding Robertson-Walker universe is far from complete (There are "known exact solutions with each [property] ... there is no known solution that possesses all [of them]"), they're just hoping to find one.
It is not at all clear to me that they have a strong idea about no-hair in their compact objects' causal structure. (I guess totally wildly that it will sensitively depend on the details of the embedding. See Visser 2014 <https://arxiv.org/abs/1407.7295>).
Also, it strikes me that their entire idea is to avoid strong gravity in the interior of collapsed stars and in particular avoid the singularity, so one should really think of this as an anti-quantum-gravity approach to black holes, or at least an approach that might evade perturbative non-renormalizability.
No extra dimensions, no boundaries, nothing special in the stress-energy tensor, mute on the subject of entropy (which in any case should be thought about in comparison with the huuuuuuuge entropy from the expanding space. Expansion is after all the focus of the topic paper, and so it's rather distant from anti-de Sitter ideas).
I just know the basic argument that if black holes are simple, then going from a complex thermodynamic arrangement without a black hole to one with a black hole would represent a spontaneous reduction in entropy. And therefore theories where black holes have a lot of entropy are of interest. While this was originally an argument for string theory, it can be used to argue for other theories as well.
I can't opine on your wild guess that how much hair their model of a not-quite black hole is depends on the embedding. But if it is true, I would expect that embeddings that give it a lot of hair are going to be of more interest than the ones that give it no hair exactly for the thermodynamic reason that I gave.
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).
