Good question. There are multiple sorts of faults we need to consider.
Disclaimer: I'm not an electrician. I have a similar background as you, some academic electrical engineering type stuff (I'm an experimental physicist). I've had similar confusion and done some self-learning on this safety stuff.
I will talk about the electrical supply for a building. There are three terminals to consider. Live, Neutral, and Ground. Note that the supply provides a voltage difference between its live and neutral terminals. This means that any complete circuit must have a path from the live terminal back to the neutral terminal.
It happens to be the case that the ground terminal is physically connected to the earth. This is not reasons having to do with clearing electrical faults but rather has to do with preventing objects within the building from floating too high above earth potential. This might happen due to static buildup, lightning, or other voltage surges. I only mention this because this one feature which distinguishes (electrically) the ground terminal from the neutral terminal. However, for the rest of this post we could COMPLETELY ignore this earth connection. However, what we can't ignore is the fact that metal surfaces in the building such as metal sinks, water pipes, etc. are electrically connected to the ground terminal.
First, a fact: at the supply for our electric circuits the neutral wire is physically bonded with the earth. This is done for reasons having to do with overvoltages that can occur due to static buildup, lightning, and other sorts of voltage surges. Basically they ensure that no circuit 'floats' too far from earth potential (note this is not for clearing circuit faults, I only mention it because we will need this fact later).
Another fact: At the supply the neutral terminal is connected to the ground terminal. We will see why this is useful and critical for safety shortly.
This first type of fault I'll consider you already understand. Suppose we had a two-prong device. There is a load which has a live connection on one side and a neutral connection on the other. Suppose there is a floating chassis. Under normal operation you can touch the chassis and a pipe in the building and have no problem. However, if there is short from the live wire to the chassis then the chassis is now at high voltage relative to a pipe in the building, so touching both could cause current to flow through you. DANGER. The way this is solved in modern circuits is that you run a wire from the ground terminal to the chassis of the device. Now, if such a short occurs then high current will from the chassis along the ground wire, through the ground to neutral jumper at the supply thus completing the circuit. This high current trips a breaker on the circuit turning everything off quickly to ensure safety until the fault is corrected.
However, forget the ground wire. If you just connected the chassis directly to neutral you would get the same effect. In this case current would flow from the chassis, through the chassis-to-neutral connection, then back along the neutral back to the neutral terminal at the supply completing the circuit. It would be high current which would trip the breaker.
What you suggest in your question is essentially the same thing. In the section you quote the suggestion is to have three wires coming from the device. A live, a neutral, and a "chassis". The live and neutral are wired to live and neutral on the outlet and the chassis is also wired to neutral on the outlet. There is no other connection on the backside of the outlet. The idea is to connect neutral directly to the chassis.
Such a configuration is called a bootleg ground. It may be tempting to install a bootleg ground in a situation when, for example, an building only has two-prong outlets (because it is old) but you have modern three prong devices. You should not do this because it is dangerous as I will describe below.
In the event of a live-to-chassis short as I have described above this bootleg ground wiring WILL cause an overcurrent that will trip a breaker to clear the fault. This is good. However, equipment ground allows us to protect against another sort of fault.
Above the issue was a short circuit. The fault we consider now is an open circuit. See this diagram from the answer here: https://electronics.stackexchange.com/a/388134/180824
See this diagram.
There are two possibilities with a bootleg ground. First, imagine it is a situation like in the diagram above where the two circuits on the right do not have any equipment ground wire running to them. Then, in the event that the neutral wire is cut as in the third device then, because of the neutral-to-chassis connection made at the bootleg ground, the chassis floats up to the live voltage. Now if someone touches the chassis and a pipe in the building they will complete a circuit and dangerous current will flow. Note that no current runs through the load after neutral wire is cut, however, if someone touches the chassis and a pipe then current does flow through the load.
The second possibility is the same as the diagram above except that the ground connections are actually completed from the supply to ALL three of the circuits. The two circuits on the right simply have the additional connection from neutral to ground. In this case, if the neutral wire is cut, the ground simply replaces the neutral wire. Current will flow through the load and return from the neutral side of the load, through the load side neutral-to-ground jumper, back along the ground wire, then along the supply side ground-to-neutral jumper back to the neutral terminal to complete the circuit. In this case current runs through the load even though the neutral wire was cut. Now what if you touch the chassis and a pipe? Well, now you are again completing a circuit providing a return path (through your body and the pipe) but you are in parallel with the direct ground wire. This isn't as bad as the previous case, in this case most of the current will run along the ground wire since it is lower resistance than you. However, if the current for the circuit is high then even the small fraction running through you could be dangerous. Also, if the ground wire was cut (or had increased resistance) in addition to the neutral wire then we would be in the previous situation as before with the bootleg ground with no true ground connection.
Very long story short:
- It is important to have an equipment ground wire running to the chassis of devices to ensure live-to-chassis SHORTS are cleared by providing a path to supply neutral to trip the circuit breaker.
- It is important that there is no connection between neutral and the chassis at the load side to ensure that the chassis can NEVER serve as part of the return path for currents. In particular such "bootleg ground" connections become especially dangerous when the resistance along the neutral/ground wires is increased due to breakages or damage to those wires.
edit: Another comment, note that under normal wiring (equipment ground bonded to neutral ONLY at the supply) that if there is a live-to-chassis short then the equipment ground line will clear the fault the circuit will trip. However, if there is a neutral-to-chassis short there will basically be no warning sign. This is somewhat acceptable because as long as the neutral and ground wires running from the outlet to the supply are in tact and the current through the load is moderate there is not substantial risk, but I still find it worth noting the lack of a warning.