Well, some neutron stars are spinning at a substantial portion of c, the record being 1/4 c. In an object the size of a city, that some serious force pulling against collapse.
The event horizon is like a waterfall, only instead of water it is causality.
To a first approximation (because time and space melting into each other in a wildly non-Euclidean environment isn’t something I can do justice to), unless the neutron star is spinning so fast it is a torus with a black-hole-sized gap in the middle, a black hole inside a neutron star will doom it.
The horizon is not anything like hard surface. You can't stack things on top of it. The near-horizon exterior gets extremely dynamical when you put lots of matter there, and the result is almost always a lot less matter there and a bigger horizon area, typically within timescales comparable to the light-crossing time of the horizon diameter. Things really do fall into (rather than onto) black holes, even if some classes of observers might struggle to witness the details and even though it is tempting for ease of calculation or ease of theorizing to treat matter as being smeared onto the horizon rather than crossing through (and/or redefining) it.
Stress-energy can find itself inside a black hole horizon with no noticeable local effects for some time. One could in principle collapse a perfectly uniform shell of ultrarelativistic matter around some victim, for example, such that the victim is inside the horizon -- that is, the victim's outgoing radio signals cannot reach the rest of the universe -- hours before the matter reaches the victim. Neutron stars require "signals" (QCD and other interactions) exchanged among its deepest components in order to stabilize the entire structure, and the system can find itself in a configuration wherein signals from region A cannot reach region B, so signals from B are wrong to stabilize region C and so on, and the result can be some combination of implosion and explosion. This doesn't fit well with the analogy of going over the edge of a waterfall.
As to your neutron-star-matter torus around a BH, it's hard to contrive even a toy a stress-energy tensor where the generating non-interacting matter doesn't stray onto plunging geodesics, and I struggle to imagine how real interacting matter with inelastic collisions could be more stable. Another salient feature of even the toy version of such system is that there would be enough moving matter in the torus for it to be subject to self-gravitation. Consequently, you're into the task of combining at least a pair of substantially different metrics in the limit where you can't just linearize, resort to perturbation theory, or drop in a Darmois-Israel junction -- that is, you're into the realm of a full numerical relativity task and a full relativistic hydrodynamics task. A taste for what this involves can be had at https://www.mpa-garching.mpg.de/181055/Core-Collapse-in-CFC_
Firstly, a torus of ultradense matter is almost certainly unphysical. Such a torus surrounding the near-horizon of a black hole is even less physically plausible.
Secondly, the waterfall analogy only (and only barely) works for an infaller who starts outside an already existing horizon. But in a black hole formed by gravitational collapse there are inside-the-horizon observers (collections of particles, essentially) who never cross from the outside of an event horizon to the inside of an event horizon: they do not fall in any meaningful sense of the word past any point of no return. They are just in a region of the universe with no horizon and suddenly in a region of the universe with a horizon, and unfortunately for them they are on the much less comfortable side of the horizon (the inside). The events that trigger the appearance of the horizon can be at a substantial distance from these unfortunate observers.
I believe I fairly explicitly said I wasn’t able to apply GR? That it was above my level? That this was a simplification? And I also believe that simplification is appropriate in this context, especially when so signalled.
I am still confused as to why you felt the need to write “The horizon is not anything like hard surface. You can't stack things on top of it.”, given I cannot see how my words could have led you to think I thought it was solid.
Actually, this leads to an important question: given that my attempt at signalling the limits of my knowledge failed, can you tell me what the right way is to signal the equivalent of “I have been comfortably following the Susskind lectures on my commute at faster than real-time, but my actual degree is not physics”?
Even classically, the spinning would pull apart the equator, but the poles would fall. You could get a disk of nuclear matter (add electrons and that's a galaxy) or a buzzing cloud where everything just misses the black hole as it orbits (like the stars in the galactic core), but if it's a solid mass with a continuous velocity field, the poles will fall in.
The largest observed angular velocity is 0.24c which sounds like a bigger issue than it is. The centripetal force from that is still significantly weaker compared to the 10^12 to 10^13 m/s^2 you get from surface gravity.
