I think the singularity theorems address it however: https://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singul...

The basic idea is that singularities are a very general feature of general relativity, not confined to specific solutions to the field equations but sort of inevitable for a large number of initial conditions. So while the standard black hole solutions are highly idealized, we have good theoretical reason to believe that singularities exist in non-trivial solutions to the Einstein Field Equations.

From the perspective of a particle falling into a black hole, the singularity will be reached in finite time.

There are a number of theories on black holes not being singularities.

PBS space time goes into a few of them

Instead, most physicists believe that a theory of quantum gravity (a theory which accounts for gravitational interactions between quantum particles in general) would slightly modify the equations of GR to arrive at a finite solution for the interior of the black hole. Of course, such a theory eludes us so far.

There's a fun thought experiment to be had here. We're all familiar with the concept of "going back in time". Even though the math doesn't check out and our imaginations of what that looks like are totally impossible, it's something we can (and often like to) imagine.

Now, what happens when you move backwards not in time, but in space? And I don't mean changing your direction 180 degrees, but moving backwards within the confines of space. So, if your present coordinates are (X, Y, Z), you move "backwards in space" 5 meters such that no matter what your direction of motion, if you move "forwards in space" by 5 meters you end up right where you started, at (X, Y, Z). You're basically creating an inverted bubble around (X, Y, Z) and making it so that all possible paths lead "outward" toward (X, Y, Z). Any point that is A meters away from (X, Y, Z) is actually A+5 meters away from you.

In a 2-D world this can be modeled with a third dimension. We all live on a flat plain but someone on a 5-meter hill is actually farther away than their 2-dimensional coordinates would imply. You're able to move away from a point in 2-D space without changing your 2-D coordinates (because you're actually moving in another dimension).

For 3D space, this extra dimension is time itself. The only way to move "backwards" in a comprehensible sense within 3 dimensions is by moving backwards within time. So, I'm not actually moving 5 meters "backwards" within space but moving backwards in time such that I will appear at (X, Y, Z) in 3-D space at precisely the same time it would take me to move forward 5 meters. I'm blipped out of space until such time has passed to allow me to reach (X, Y, Z) again, so I've successfully "moved backwards in space" by 5 meters. So, if I move "backwards in space", then I'm really putting (X, Y, Z) in front of me time-wise.

So, what's the center of a black hole? We can imagine it has coordinates (X, Y, Z). And let's pretend it's actually moving backwards into space, which we now conceptualize as putting its "present" coordinates forward in time. But, it's actually moving backwards in space (i.e. moving universally farther from every other point in space) at a speed faster than we can travel. The center of the black hole is always in the future and even as we move closer to the center, it's still infinitely far away. As we cross the event horizon, we ride the current and start moving faster than the universal speed limit, cutting ourselves off from ever interacting with the outside universe again. However, we're still never going to be able to reach the center, and indeed the center of the black hole has still not happened from our point of reference (and still never will).