Firstly, note that in some parts of the world, neutral and ground are NOT connected at the house, but back at the substation (transformer) so there may be a few volts on neutral in those systems.
But to the question : Consider if you did just connect chassis to neutral.
Then what happens if part of the house wiring fails open circuit?
1) Live (Hot) fails ... appliance stops working safely.
2) Neutral fails ... appliance stops working - with the chassis live!
This is not good.
Connect the chassis to earth and what happens when a wire fails?
1) Live (Hot) ... appliance stops working safely.
2) Neutral ... appliance stops working safely. If a current path develops between live and earth (perhaps you are trying to fix it) 10 or 20ma will trip a modern breaker disconnecting the supply.
3) Earth ... Nothing happens unless something else goes wrong. The earth failure will be caught at your next scheduled safety test. Right?
In a nutshell
Electricity is not supposed to flow through ground stakes in normal conditions. It doesn't mean its resistance is high, it's actually surprisingly small. That branch of the circuit is simply not closed normally.
In details
A ground is a reference point. You could litterally take any net in your circuit which is supposed to stay at a steady voltage and call it ground. After all voltage sources create a difference of potential (called a voltage) between two nets, regardless of what their potentials are - if they're both fixed externally, there will be a conflict and bad things, but if one of them is fixed the other potential will change accordingly. Generally the ground is taken such that we work with positive supplies predominantly, e.g. ground on the - terminal of a rectifier bridge. It doesn't mean all the current flow through that, it's only a reference.
The Earth has mainly a person protection role. No current is supposed to flow in the Earth because the actual supply circuit is isolated from the Earth, however what if this isolation is compromised (wires eaten by rabbits, children shoving their fingers in sockets...)? Everyone is indirectly connected to Earth (no isolation is perfect), which means that that circuit will now be closed and the only thing that will limit the current going through whatever is closing the circuit (e.g. people) is its internal resistance. Depending on the environment, that resistance can be sufficiently low to kill someone; refer to this thread about what voltages are considered safe. To prevent that, every enclosure is connected to Earth (a Earth-R-Earth circuit has a near-0A current), and the electric supply has a residual current device that compares the current going in and out, and cuts off the supply if there is a leak (through Earth).
The Earth is used for an equi[reference]potential supply The electricity provider needs to protect its people too, so the upstream supply is also referenced to Earth. Just like everywhere else. So what happens if the Earth is not a good conductor and its potential is not homogeneous? Users could be in contact with 2 different Earths, which can be a high difference of potential (=voltage). Thankfully, moist in dirt and water patches are good conductors, but above all the equivalent cross section of this fictive conductor is massive. Except during short upsets such as lightning, it has an excellent homogeneity in potential. Why use another conductor for ground which will use more copper and actually be less effective if we can use what's under our feet?
The Earth is also useful as a protection against lightning: lightning is just like any dielectric/isolator breakdown, it occurs where the resistance between the charged cloud and the Earth is minimal (see this amazing GIF). High trees, towers etc., and we can't risk relying on luck alone so highly conductive spikes are used to attract lightning, and the Earth is used to dissipate that energy. Loosely said. Normally lightning has enough current flowing to create through Earth and across human legs a voltage high enough to kill them, so it is spread out more evenly.
As usual, I'll warmly welcome anyone correcting me if not accurate.
Best Answer
As others have mentioned primary reason is earth/ground line is a backup line separate from the neutral line to provide two fault protection.
That is, more than one thing needs to fail. For a grounded chassis device the chassis can become live only if two (or three faults occur).
Two fault case: Earth connection broken AND Live wire shorted to chassis.
Three fault case: Earth connection broken AND Neutral wire shorted to chassis AND Live-Neutral reversed at outlet.
There is actually a single fault case too though.. "Earth pin is wired to live at the outlet..." but there is nothing you can do about that one...
Also, under normal circumstances the earth line should have zero current flowing through it. That is quite different from the neutral line that can have 10s of amps running though it which can produce significant voltage drops at various points along the conductor. The ground line "SHOULD" be at zero potential everywhere.
By the way, the wiring you show for House#1 is not legal in these parts. A separate ground is required at the residence, usually clamped onto the incoming water line if it is a metal pipe, or to a long ground spike. Neutral may or may not be tied to this ground at the fuse box. This provides better local protection as opposed to relying on the electric company and the integrity of the service which may get disrupted (torn down) during a storm.
Yes, it is unfortunately very easy and common for bad wiring to exist where line and neutral are reversed at the outlet. Most appliances don't care, but if you plug your device with the neutral tied to the chassis into such an outlet.. well you can figure out the rest.
Further there is the broken neutral line scenario which can also be quite lethal. Consider the drawing below.
simulate this circuit – Schematic created using CircuitLab
Lets assume appliance 4 used the neutral to ground the chassis. Since neutral does not actually go anywhere the chassis is really floating.
However, what happens when you turn on appliance 5. The neutral line coming out of appliance 5 is then pulled up to the live rail. Appliance 4s chassis will also become live. Appliance 5 could be in another room on the other side of the house...
Standards vary around the world and age makes a big difference. People seem to forget that residential electricity is a relatively new phenomenon. In the early days things were A LOT more dangerous. It is only through time that we have developed more common and safer standards. Though, as I hinted, some parts of the world are still rather lagging due to the costs to replace everything... Canada/USA for instance, where you can still stick a plug into an outlet while touching the live pin....