The zone system was designed to help translate the Scene Brightness Range (SBR) you want to print from the real world to what will print easily or "straight" onto say grade 2 paper.
Adams would have used the word visualize where I used translate. When looking at the scene Adams would visualize (decide) what he wanted pure black-zone 0, shadow with detail-zone III, white with detail-zone 8, and what he wanted pure white, zone X.
The black point Adams would choose in the scene (zone-0) translates to black on paper (zone-0), each zone from the scene translates in turn directly to the corresponding zone on paper. That is the essence of the zone system.
With regard to negatives, they almost always have more "range" or latitude or usable details (whatever you want to call it) out side the printable "zones" or straight print range. To say this in mathematic terms, the printable range (all the zones) are a subset of the film's range. It is very possible for a film to have twice the range than it needs to cover the zones.
Two side notes:
We can get at the extra info/extra film range by using burn or dodge techniques.
And the classic placement of a Caucasian face tone in Zone VI isn't a given. For a dark scene you may want to print darker, for a bright scene, brighter.