At the very best it could stop SSTO to orbit suck so much. But it seems SpaceX is far enough that it is not possible to get close to the efficiency with an autophage rocket. It maybe was possible as long as first stage was always destined to crash, but when it can land safely the entire efficiency equation got turned on its head.
Would that kind of rocket be usable for deep space (basically -- improve thrust to weight by eliminating the weight of the fuselage as you thrust)? I also don't think so. For space, much higher specific impulse engines exist than any kind of fuselage material that can burn itself.
So I just don't see any use of this.