The kind of "no singularity" model Kerr discusses--where you have an object inside the inner horizon that replaces the ring singularity--is not actually viable. First, most physicists believe the inner horizon itself is not viable, because it is unstable against small perturbations, which any real hole will have (because there is always
something falling in, even if it's just the cosmic background radiation). Second, it's not actually possible to replace the
entire ring singularity with a non-singular object.
However, there is a different type of "no singularity" model in the literature, called a "Bardeen black hole" (among other things), in which there is not only no singularity, but no event horizon. What appears to us as an event horizon is actually just an apparent horizon--a surface where, locally, light cannot escape, but which will not stay that way forever; eventually it will evaporate away (by a process similar to Hawking radiation), and everything that was trapped inside will come back out (though in hugely scrambled form). So there is nowhere in this spacetime that can't send light signals to infinity; it just takes a really long time for the signals to get out from some parts of it.
However, there is a different type of "no singularity" model in the literature, called a "Bardeen black hole" (among other things), in which there is not only no singularity, but no event horizon. What appears to us as an event horizon is actually just an apparent horizon--a surface where, locally, light cannot escape, but which will not stay that way forever; eventually it will evaporate away (by a process similar to Hawking radiation), and everything that was trapped inside will come back out (though in hugely scrambled form). So there is nowhere in this spacetime that can't send light signals to infinity; it just takes a really long time for the signals to get out from some parts of it.