I know that the answer to my question is "Yes, it is!". The ground connection is needed to avoid a person touching a metallic part of faulty equipment to be electrically shocked. So the person acts as the conductor closing the circuit between the high voltage metallic part and the ground (at zero volt by definition). The grounding connection hence offers an alternative path for the current to flow, and this happens because somewhere along the power distribution system, the neutral wire N is connected to a metallic rod planted into the ground. My question is: what if this neutral wire - ground connection is removed, as illustrated in the picture? This would make the ground wire in home electric sockets unnecessary, since a person touching the metallic case of a faulty equipment doesn't close the circuit, because there is not the circuit including the ground. So, as I understand it, this would work (maybe) if the power distribution system is not grounded anywhere. Is this possible to realize a totally floating-from-ground power distribution system? I am pretty sure the answer is no, but I don't know why... so, why is not possible??? Thank you.
What you are not considering is that an earthed chassis or metal box containing a "rogue" live connection will be self-revealing should a fault occur that causes live to touch the earthed metal. It will blow a fuse and within a few milliseconds the problem is made safe and revealed.
For an isolated AC source (such as the secondary of an isolation transformer), there is nothing to reveal that live may have come into contact with the metal box. You can touch it and maybe receive a small tingle but, it isn't clearly self-revealing.
Fault 1 has occurred and it is not self-revealing...
Some time later, another fault occurs that can be anywhere on your street and in any of the houses on that street. That fault causes neutral to become connected to earth.
Fault 2 won't self-reveal either because it won't blow a fuse: -
Then you touch the unearthed metal box or chassis: -
The magenta arrows show the flow of electrocution current
Now, the metal box is a death trap (because it isn't earthed) and is awaiting the next unlucky person to come along and touch it. Nothing has revealed itself that tells you a dangerous situation has arisen.
The first "hint" (if you survive the hint) is electrocution.
In addition to Andy's answer, a floating supply may require dual circuit-breakers on each circuit to trip both lines as you can't predict which phase will be "neutralized" by the first fault.
Deliberately neutralising one line avoids that risk and therefore only the live wires require fuse or circuit-breaker protection.
The earthing system shown is the TT system followed in Italy with a three phase supply voltage of 400 V and single phase 230 V.
In this system, the responsibility of transformer neutral earthing rests with the utility service provider while that of metal enclosure earthing is with the consumer.
The system permits use of circuit breakers (MCBs) and ground fault circuit interrupters (GFCIs) to clear line-to-line, line-to-neutral and line-to-earth faults.
The most probable location of a line-to-earth fault would be in a metal enclosure. Should the neutral and metal enclosures not be earthed, no tripping would occur to clear such a fault and it would remain unnoticed. The natural outcome would be electrocution due to contact with the live metal enclosure and earth, should another line or the neutral get simultaneously earthed elsewhere.
In a three phase system, without humans (or valuable animals/equipment), and no conductor breaks, there is no need for that redundant 'Neutral', or 'Earth' connection.
The problems come when there are faults, Humans, and valuable animals/equipment present that need protecting from the electrical power (fire), body currents (shock/death), and over voltage (equipment).
The earth, and bits of metal stuck into it, provide a good conduction path for humans who touch the faulty system and the 'earth'.
All the fuss is about trying to ensure that humans (and valuable animals) that are normally touching earth don't also touch faulty equipment that has voltages that can induce shock currents. And also that the fault currents (in the wires) are high enough to blow the fuses (circuit breakers) which needs a low impedance to, and through, the Earth.
Hence we make sure there is sufficient insulation and distance so people can't touch the live circuits, that the circuits are fused and earthed (back up return connection), and also nowadays that we also have leakage (residual current or ground fault) protection devices.
Earthing isn't actually complicated, but too many different aspects are all swept under the same carpet (E.g. damage of supply equipment by overheating, death by fires, death by electrical shock, equipment damage by voltage transients, to identify a few of the fault protection measures in the modern supply).