Also the EGC does have more purposes than simply clearing fault current - for example carrying away leakage current from the chassis of something with a switching power supply where the output can't be completely isolated to meet emissions. Ever been shocked by a laptop with only a two-prong AC adapter?
IMO this whole subject is a minefield of disagreements due to terminology, when really all questions are answered by drawing out the schematic of a typical electrical system and looking at the loops (circuits). For instance in the typical ground fault, the fault current returns to the distribution transformer in parallel through all of: your service's neutral, your grounding rod, your neighbors grounding rods/service neutrals, and your other leg of the split phase via turned-on devices. This seems like a lot of unrelated details to memorize until one draws it out.
EGC is bonded to the panel enclosure at the ground bus. The neutral bus is also bonded to the enclosure with a ground screw. So, the EGC is directly connected to the neutral bus and a fault between hot and a bonded enclosure is routed via the EGC to neutral, tripping the breaker.
If you don't bond neutral and ground at the main, it's still bound at the transformer.
> then the current will flow through the ungrounded conductor,
No, it will flow through the grounded conductor.
> back to the source (usually a transformer), across the windings, and eventually back to the breaker that controls the circuit
The flow of power is the same via ground as it is via neutral.
> This will build up enough current (usually VERY fast, like milliseconds)
There is no such thing as "build up enough current", current does not "build up".
The milliseconds has to do with built in time delays at breakers, it's not a function of the electricity.
> The Earth has nothing to do with this
The actual Earth is used as a conductor, so yes, it definitely has something to do with this.
That would be a pretty good trick. The transformer is outside my house on a pole. Two wires run to it, the hot and the neutral. The "ground" doesn't leave the house. (Other wiring schemes exist, but this is standard USA residential.)
[EDIT: brainfart, see helpful correction below. still no "ground" at the pole...]
Yes there is. The neutral is bonded to ground, via an actual metal pole in the ground.
In your house is the same thing: The neutral (and the ground) and attached to a metal pole in the ground.
And the earth itself completes the circuit.
Standard US residential is 240V split phase, with two hot and one neutral conductor from the transformer. So three wires, minimum. Unless you have a very old feed.
The entire purpose of the EGC is to bond all normally non-current carrying conductive parts together to provide a ground-fault current path back to the source of the circuit. If the EGC is not connected to some non-current carrying conductive part, then it is not installed correctly, is against code, and is a safety hazard. If the EGC is bonded to everything properly and there is a fault (say an ungrounded wire touches a metal box), then the current will flow through the ungrounded conductor, through the metal box, through the EGC (which is bonded to the box), back to the source (usually a transformer), across the windings, and eventually back to the breaker that controls the circuit. This will build up enough current (usually VERY fast, like milliseconds) and the breaker will open. The Earth has nothing to do with this.