Putting the lightmeter under the chin makes sense to me because:
a) It is incident light what's being measured, and the little sphere is to be located as close as posible to the surface being iluminated (i.e. the face's skin). Putting the lightmeter at the same distance from the light source is crucial as light intensity varies with distance to the source. The key point here is "incident". Both the dome of the meter and the chin are exposed to the same source and at the same distance from it.
b) You can not put the hideous thing, along with your hand right in front of a person's face, It is plain rude and will make the person uncomfortable, so later when actually shutting it would be more difficult for the portraited person to achieve a natural, confident and relaxed look. (They may be wondering when the next personal space violation is going to take place). In this regard I have even seen photographers hand the meter to the model and politely ask them to put it right besides their face, thus avoiding intrusion.
Reinforcing point a): It's not absolutely necesary to put the light meter under the person's chin, you can put it right at the side, trying to locate the sphere near the eye of the person. Some photographers I have worked with, actually take several measures, one by each eye and other under the chin. With this they measure main light (in one side of the face), fill light (in the other side) and the sum of both (Under the chin, where both light sources are "visible").
If clothing is particularly important for the photo (like a fashion shoot or for a clothes/accesories catalog) or if skin is showing besides the face, you may need to measure other areas as well. A typical example may be if the portraited person is showing some amount of cleavage, you would like to balance that with the face, or at least make sure it won't be brighter than the face.
If you where to measure reflected light instead, then you would remove the white dome and turn the light meter around (i.e. facing the subject, not the lightsource). In this procedure, the light reaching the sensor in the meter is not coming from the souce, intstead it's coming from the person's skin. In such case, it's more likely you would measure also the cheek bone, forehead and nose, because you would be making sure of wich contrast level you would produce between those areas.