The problem with watching for it to "run true" is, that is a judgment call, not an event.
The wobble allows you to see your progress and slow down to just barely moving the table at the point of kickout. Yes, you have gone too far but you have just barely gone too far, 10ths? The "run true" method works as long as your judgment of "true" is in the few 10ths range.
Machining has lots of built in errors and there are always ways to come closer to the magic "perfect." But never get there. A mill, even a high end CNC will have errors. Grinders can come closer to the illusive perfect. As long as I'm on a rant about what is essentially the argument over "good enough," what about the surface texture your method of choice leaves? Most measuring devices use the high spots, but they can compress or rub off.
I think my edgefinder is an import, Brown & Sharp. It seems very nicely made and repeats within a few 10ths running at about 900 rpm. My hands are likely not steady enough to do better no matter how good of a finder I use. I'll bet many of the cheaper finders can do as well. Just do a test of repeatability to satisfy yourself.