I'm going to suggest a small change in word usage.
"over expose" tends to imply too much exposure.
"under expose" tends to imply too little exposure.
Both tend to imply a mistake.
For clarity, it is really helpful in these sorts of discussions to instead discuss actions - we decide to "increase exposure" or to "decrease exposure" and we usually speak in terms of numbers of stops.
And, when we compare the light reflecting off of various parts of a scene, we talk in terms of stops as well.
The reason to use a grey card is that it supplies a repeatable reference. The reflectivity of an 18% card is approximately the middle of what we encounter in real life. So if everything is calibrated correctly, if we meter off of a grey card (using the proper technique) then the image of any 18% tone in a scene will be recorded on the negative to a density that should print as a middle grey tone in a print, and the darker and lighter tones in the scene should print appropriately as well.
The reason we use a meter to gather information from the shadows, is twofold:
1) negative film has less room for error on the shadow side, and more room for flexibility on the highlight side. Accordingly, we meter the shadows to be sure we capture the important information there; and
2) not all scenes have a 18% reflectance in the middle. Some scenes are such that you want the middle to be darker. Other scenes are such that you want the middle to be lighter.
It is important to understand that with respect to the zone system, you don't meter to find the "darkest thing with detail". You meter the darkest thing that you want to have detail in your final output (usually a print).
Let me give you an example. I take lots of photographs in forested areas. There are lots of scenes there where there are dark shadows that I'm quite happy to render dark and free of detail in my prints.
If I am using zone system principles when metering, I'm visualizing the result I want in the print and then measuring the light reflecting back from the various parts of the scene. Those measurements tell me that if I adjust the camera exposure to place a certain part of the scene (the part where detailed shadows matter to me) at one level, the other parts will fall at other levels. My experience then tells me how the resulting negative will print in a straight print, and will give me a good idea how it may print if I also apply the printing controls available to me.
The zone system is a way of systemizing the following:
1) visualizing first, and then
2) accomplishing the vision.
It all comes back to the final output - the print.