As you seem to mention, the primary purpose of the ground wire is to hold the OUTSIDE case or enclosure at the safe, earth potential. That outside enclosure is what people touch, what they handle the device by. And yes, as you guessed, if a live conductor does touch that outside enclosure, then the fuse or circuit breaker in the service panel should instantly blow or trip, removing all power from the device. And keeping the user safe. Fire prevention is also tied up in here. This type of protection comes from a time when most electric devices had metal outer enclosures, not the plastic of today. It is still applicable to most commercial and industrial equipment where metal is still the primary material.
The problem with using the neutral for double duty, for the return path for the current and for a safety ground is if the neutral wire should break or somehow become disconnected between the device and the service panel. Then the outside enclosure that is still tied to the neutral will become energized at or somewhat below the line potential. And then the user can/will be shocked and possibly injured or killed. The path for the line Voltage would then be from the hot wire, through the internal circuitry of the device, to the neutral wire inside the device, then to the outer metal enclosure, and finally to the person holding the device. And through that person to any ground he/she may be in contact with. The degree of the shock would be determined by the resistance through the device's internal circuitry and how well that ground contact may be. It could vary from a tickle to a lethal shock.
According to the US code the ground wire can be eliminated if the outer shell of the device is DOUBLE insulated from any current carrying conductors. Thus, some devices like lamps or hand held power tools are equipped with a two conductor cord. In theory, at least two separate insulators need to fail before the user can get shocked. In these devices, if the outer shell is metal, then it can NOT be connected to the neutral or hot conductors or any internal parts connected to either of them. So a break in the neutral, as I described above, will only cause the device to stop working. It will not result in any shock hazard.
The electric codes must account for many possible failure scenarios, not just the simplest or most obvious ones. Oh, and according to some sources, the fuses or circuit breakers are originally there for prevention of fires when wires are shorted, not for the protection of human lives. But protection of humans has become one of their purposes today. I tend to believe this.