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.
Is there an "easy" example showing the fact that connecting the neutral to chassis is a bad idea?
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...
Imagine one lives in a house and there is no earth in the outlet. I have seen such apartments and never understood how they are allowed to wire houses like that.
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....