Question for anyone who has used one of these analog measuring devices: the indicator seems to go all the way around before the camera zooms in to read the indicated value. Is this video actually showing the accuracy it is claiming?
Yes. The sphere at the tip has a certain radius, and the indicator will show zero (again) when the sphere has been deflected by its radius (i.e. the contact point is exactly at the center line). When out of contact, it's essentially telling you that you're missing at least a whole millimeter to the point where you should be.
Often there is a second needle indicating which of these situations you're in, but I assume it's not considered necessary because if you're 1mm off, the situation is (in the contexts in which these devices are used) very obvious.