> Why should it be hitting the singularity at all?
Because the singularity is in the future of every event inside the horizon. It's a moment of time, not a place in space. It can't be avoided because tomorrow can't be avoided.
> Why can't particles have a stable orbit just under the event horizon? Is it because the orbit speed would exceed the speed of light?
That's one way of looking at it, yes.
> why singularity at all?
Because that's what the math of GR predicts.
> Neutron stars compress to a certain density and then stop compacting any further.
Neutron stars have a maximum mass limit. So do white dwarfs and every other kind of stable compact object. Anything over that maximum mass limit has to collapse into a black hole.
> Can't it be a very dense object where compacting force is compensated by matter density limit?
Not for objects above the maximum mass limit. See above.
> Neutron stars have a maximum mass limit. So do white dwarfs and every other kind of stable compact object. Anything over that maximum mass limit has to collapse into a black hole.
See, this is the crux. From what I read in GR theory, once the neutron star gets over mass limit, the gravity becomes that big that light can't escape it any longer. OK. Light can't escape. Buy why would a matter suddenly work radically differently under the event horizon?
Imagine an asteroid, which has an escape velocity of 1m/s, so it's inhabitants can launch pinballs to outer space with a simple catapult. This asteroid eventually collides with another one, with an escape velocity of 2m/s, so pinballs wouldn't leave the body's gravity field. Nothing really changes on the asteroid, all physics processes work in the same regular way.
Why should it be that different with neutron star vs black hole? NS with ~3 solar masses has radius ~10km, and is a giant lump of neutrons packed together. To me it looks like BH should be the very similar lump of neutrons, just so heavy that light can no longer escape. So what? It should still be a spherical lump of neutrons underneath, just denser.
> From what I read in GR theory, once the neutron star gets over mass limit, the gravity becomes that big that light can't escape it any longer.
No, that's not what happens. What happens is that if a neutron star just under the mass limit gains some mass, it collapses into a black hole. It does not just change a little bit. It changes a lot.
There is no such thing as a static object that is "just short" of being a black hole. A theorem called Buchdahl's Theorem says that no static object can be smaller than 9/8 of the Schwarzschild radius for its mass, i.e., 9/8 of the size of a black hole of the same mass. That means there is a finite gap in size between any static object and a black hole. So any transition from some static object, whether it's a neutron star, white dwarf, or anything else, to a black hole can't be a small continuous transition.
> why would a matter suddenly work radically differently under the event horizon?
It's not "matter" that "works differently", it's the geometry of spacetime itself. The geometry of spacetime inside the event horizon is simply different from what you are used to, and has different properties than those your intuition is familiar with. One of those properties is having a singularity a finite time in the future.
> To me it looks like BH should be the very similar lump of neutrons, just so heavy that light can no longer escape. So what? It should still be a spherical lump of neutrons underneath, just denser.
No, that's not what a black hole is at all. A black hole is vacuum inside. It is not an ordinary object made of matter. It is made of very unusual and counterintuitive spacetime geometry. Nothing else.
As for the matter that originally collapsed to form the black hole, it continued to collapse until it reached zero size and formed the singularity. It isn't there any more. While it was collapsing, its physics was the same as for any other body of collapsing matter; nothing about that changed when it fell below the event horizon. But the black hole is not made of the matter that collapsed.
The "black hole" concept you are describing would make sense in Newtonian physics (and indeed John Michell in the 1700s proposed a similar model of a Newtonian gravitating body with escape velocity greater than the speed of light that worked like this), but it is not possible in GR. GR is not the same as Newtonian physics.
Because the singularity is in the future of every event inside the horizon. It's a moment of time, not a place in space. It can't be avoided because tomorrow can't be avoided.
> Why can't particles have a stable orbit just under the event horizon? Is it because the orbit speed would exceed the speed of light?
That's one way of looking at it, yes.
> why singularity at all?
Because that's what the math of GR predicts.
> Neutron stars compress to a certain density and then stop compacting any further.
Neutron stars have a maximum mass limit. So do white dwarfs and every other kind of stable compact object. Anything over that maximum mass limit has to collapse into a black hole.
> Can't it be a very dense object where compacting force is compensated by matter density limit?
Not for objects above the maximum mass limit. See above.