It's use is based on certain things being true. One is that the chuck jaws themselves being capable of holding the work centered, yet being loose enough to be pushed around. On a tight chuck with jaws that are simply not running true, it is of limited or no real value. Or with a 4 jaw, if the work is not indicated true, you can't push it into true running with this tool. If you indicate the work right at the jaws, and then move out several inches and find the work running out, then perhaps you can improve things a little with this tool, but the fact that the work is running out at that point indicates a problem with the jaws. I'm not saying that any 4 jaw will hold work perfectly straight way out from the jaws even if new, but in any case, the work should be tight, and not all that easy to push around, with any tool.
For pushing the face of a workpiece into true, yes, this tool may help, but not so much for axial runout.