Even though a black hole has a Scwarzschild radius that indicates a finite small distance to the center of the hole, the distance traveled by an infalling particle seems a lot bigger than the Schwarzschild radius due to the extreme curvature of spacetime. An infinite curvature even seems to imply an infinite distance. Two particles falling into the hole, one after another, end up spatially separated. This holds true for all particles that fall into it.
This seems to imply that the distance traveled is actually infinite. I mean, if particles end up spatially separated because of spaghettification, and they all fall the same distance to the singularity, only an infinite distance will do, so it seems. What (if) is the flaw in my reasoning?
EDIT
I'm not sure if the linked question is a duplicate as it asks about the time it takes to freely fall to the singularity. I'm asking about the distance traveled, and I read in a comment by @MichaelSeifert that there is a good explanation for why the concept of distance isn't well-defined in this case. I can't see why that is so. What if you, the freely falling indestructible observer, have an infinite rope, one end of which you attach somehow just above the horizon? Can't you see how much rope has rolled off from your device when you hit the singularity?
