Let's think of it this way.............................
We can just call it "The Green Wire".
The green wire has to be connected to the enclosure so that it can carry voltage/current back to the power pole if a hot wire contacts the enclosure. The breaker needs a ton of current to work FAST. The green wire allows a TON of current to flow back to the pole. This all happens in a millisecond..........hopefully you're not in contact with the enclosure during that millisecond.
To carry that large amount of current, we need a correctly sized green wire
https://conduit.site/tables/table-250.122.php
The correct wire size allows a rush of current to flow from the pole, through the hot wire, through the breaker, through the green wire.........back to the pole. That rush of current trips the breaker.
Forget about anything to do with a ground rod. The ground rod is for lightning(to keep it simple). It has nothing to do with tripping the breaker.
If we don't have the green wire.........where does that hot current go? If you touch the enclosure, it goes through you, through the earth, up the ground rod(in a grounded system), directly to the neutral.........and back to the power company. You're the circuit.
Terminology is the real problem.....................a grounding wire is a grounding wire if it's connected to a ground rod...........but it's also a path back to the pole if it's connected to the neutral at the service disconnect. It serves 2 purposes........................either clear a fault, or dissipate DC current to the Earth from a lightning strike.