One other point is that that it depends on whether you are trying to measure runout accurately or just centre a cutter or piece of work. For example, if I'm setting up something in a 4-jaw chuck, I'm only looking for zero needle movement and actual value doesn't matter too much, so so bit of misalignment is ok. But if you are measuring actual runout...
Is the problem that you can't see the point of contact, so you can't see if the plunger axis is aligned with the shaft you are measuring?