One possible explanation is that I am using an awl to draw the radius line, and depending on where your hand is when scratching along the radius marker of the centering tool, the tip of the awl can be either directly below the diagonal marker on the tool, or many hundredths of an inch to the one side or the other, depending on whether the awl is exactly vertical or at an angle. Unfortunately, because there is nothing that clamps the centering tool onto the rod, and I am somewhat disabled in only having two hands (instead of the far more useful three or four hands), you may not get exactly the same angle with the awl each time. What this means is that if you draw several radius lines, some of them are going to be more remote from the true radius than others.
One solution to improve the accuracy of my center point was to take the micrometer, measure the maximum diameter of the rod, and move the micrometer to half of that diameter. Then I put one jaw on the edge of the rod, and pivoted the other side so that its jaw scribed a line through the center of the rod. By doing this twice, the intersection of these two circles with the radius lines gave me something pretty decent--perhaps as accurate as .01" to .03". That's better, but not as good as I would like.
I am thinking that what I may want to do is take a piece of aluminum square tube 5" to 6" long, and drill a hole in each leg, then drill and tap into the sides of the centering tool so that I can screw the right angle onto the centering tool. Then I can drill and tap some 1/4"-20 holes in the square tube walls to lock the rod that I want to center in position. Then I can put the whole assembly in a vise, and use the awl to scribe the radius lines without having to hold the centering tool on the end of rod with one hand, while holding the rod firmly with the second hand, and scribing with my non-existent third hand.
Or perhaps I will take the next step: start with a square tube large enough to handle the rod, drill and tap the 1/4"-20 holes to hold the rod in position, and machine an end plate perhaps 1/8" thick that looks like this:
I would cut two 45 degree slots that were just wide enough to accept the tip of the awl. They would be in immediate contact with the rod, so it would be almost impossible to get anything but a directly vertical relationship between the awl point and the rod. This should make it possible to get a center that is thousandths of an inch accurate, not hundredths, and with very little effort.
In addition, by having two radius lines, there is no need to rotate the rod to get two crossing radii. You could still rotate it after drawing these two lines, and quickly scribe two more crossing radii lines, improving accuracy, but it wouldn't be necessary.
Of course, like the Groz centering tool, it would work with any polygon with an even number of sides: squares, rectangles, hexagons, octagons. I don't have much occasion to need to center pentagons or heptagons, and nonagons and undecagons are, shall we say, not common machining problems, except for the machine shop in Flatland.