It is consistent to say that anyone at that point all of a sudden gets an extra "coordinate system" they can interact with, but they can never leave the point i

It is consistent to say that anyone at that point all of a sudden gets an extra "coordinate system" they can interact with, but they can never leave the point in the outer coordinate system. That would be 'every singularity contains a new universe'. It's not a very scientific theory, since you cannot report back after attempting to test its predictions. Though the predictions can be tested empirically!