So if singularities don't exist, then some other weird object must that naively looks like one.
You could say that we do observe a singularity, not in the centre of the black-hole but in its event horizon. But technically that's just an infinity in the maths not a physical singularity, in the sense that if you were there it would just seem like normal space.
I am assuming that these are the only way to describe anything within an event horizon.
The no-hair conjecture says that a Kerr-Newman black hole (KN BH: stationary, eternal) has only position, linear momentum, angular momentum, and electromagnetic charge.
3 components of linear momentum and three of position can be removed by keeping the KN BH at the origin of a system of coordinates. The KN BH doesn't evolve with time so we can remove two related time components as well.
This leaves us with 3 components of angular momentum ("spin"), electromagnetic charge (because by definition the KN BH is immersed in an electromagnetic field), and mass.
Schwarzschild BHs are a special case of KN BH where spin and electromagnetic charge vanish.
But we could add other fields with charges, and make those charges more complicated thanks to interactions among the matter fields. That's not a Kerr-Newman black hole any more, though. In practice the other standard model fields don't really make much difference: the charges will tend to neutralize before gravitation is relevant, and won't build up around the BH itself. KN BHs are in that sense electromagnetically quasi-neutral.
Things falling into a KN BH cause a perturbation that decays quickly away, changing the mass, and possibly spin, of the BH held at the coordinate origin. A change in electromagnetic charge will probably "reach out" and capture a charged particle in order to neutralize. Electromagnetic attraction is much stronger than gravitational attraction, while also being long range. However neutralization is not necessarily instant (the closest proton might be several light-minutes away, for example) or completely matched (oops, two protons are electromagnetically nudged into an infalling trajectory in response to the small negative charge, so when the second one arrives there'll be a small positive charge for a bit until a further electron is pulled in...). So quasi-neutral.
These parameters are only about the horizon. It says nothing about whether the mass was a bunch of individual protons and electrons or a bunch of neutral hydrogen or a bunch of heavier molecules. (The same "says nothing" also exists classically without reference to particles: start with a Schwarzschild BH and drop in a uniform spherical shell of mass M or two concentric uniform spherical shells of mass M/2 each, and after some "balding" time we cannot tell whether our increased-mass Schwarzschild BH had one or two shells dropped into it).
More prosaically, "no hair" is a statement about the stability of black holes in the face of small perturbations. If you throw something (a star, a big gravitational wave) into a black hole and wait a bit, does the resulting configuration still look like a black hole in the sense that the trajectories one grinds out of the Kerr-Newman metric (with adjusted mass, spin and charge parameters) accurately represent the trajectories around the new configuration?
There is a lot of literature on black hole stability justifying a "yes" answer.
And this Schwarzschild density is comparable to the mass density of neutron stars (or atomic nuclei) for small black holes, and can be as low as the density of water (!!) for the supermassive ones.
Naively, I would have expected this to provide a good lower bound for "largest possible mass density", but its actually just lower than neutron star density for pretty much all black holes observed so far.
See https://en.wikipedia.org/wiki/Schwarzschild_radius (=> "Schwarzschild density").
Some details:
Christodoulou & Rovelli 2014 showed that the maximal interior volume of a Schwarzschild black hole increases with time <https://arxiv.org/abs/1411.2854>. There is plenty of follow-on work in the literature for other varieties of black hole, and it is pretty generic that a black hole gets REALLY BIG inside as it gets older.
I have previously aimed HNers at DiNunno & Matzner 2008 <https://arxiv.org/abs/0801.1734> (this is also good teaching material for thinking about different systems of coordinates on a Schwarzschild BH spacetime).
One might get some intuition by thinking about pouring material into a (non-static) black hole. When do you saturate the black hole? Does it ever fill completely up? How does its outer boundary (the horizon) evolve as you pour more and more material in?
What we see of black holes isn't the singularity.