I think you are misunderstanding the tech in question. This doesn't work on all radioactive waste, just the graphite (which admittedly is a decent portion of it). You'd still want secure facilities to store what you haven't recycled.
Depleted Uranium and the stuff that comes out of depleted uranium (which I think would account for most of the spent fuel in the United States) emits beta and even shorter alpha radiation. Does that mean that the concept could work with depleted uranium?