We force it to be that way
Mains power is wired as an isolated system, with an asterisk. The asterisk came about for some very good reasons. The "safeness" of neutral is a side-effect, and an optional one.
If mains power were an isolated system (And I've run it that way, and it works), and you are grounded presumably... then it wouldn't matter if you touched pole 1 or center (I won't call it "neutral"). No current would flow. The hot and center have no relationship with earth (except through you, and with only one "wire", it's an open circuit). The system "floats".
An isolated system is exactly what you expect.
However, we build mains power to be resilient when something goes wrong. Things can go wrong with isolated systems, and one of the scariest is a transformer leak. If transformer primary leaks (even a little) into the secondary, or if there is capacitive coupling, then it de-isolates the isolated system, and "pulls it up" to thousands of volts compared to ground. Now we have a problem. In that lathe motor, coffee maker or LED light, the insulation is not rated for thousands of volts.
The equipotential bond makes the neutral
To prevent the secondary ("isolated system") from floating at high voltages, we intentionally add an equipotential bond to force a relationship to earth. You might use a transformer for the equipotential bond, e.g. in 3-phase delta (non-wild-leg) to put earth in the middle. You could also use a car battery, giving the system a 12VDC bias from earth. But usually, you use the cheapest equipotential bond available: a piece of wire. You bond one of the conductors to ground, typically "center". **Because it is bonded to earth, you label it 'Neutral'.
It really doesn't matter which supply wire you bond to neutral. Ideally you want to minimize the voltage (to earth) of the hottest hot, so the best choice is in the electrical "center" ... however, 240V wild-leg delta is an example of not doing that.
So to answer your question, neutral is cold because we made it cold.
Neutral is not quiescent; it pulses at line frequency just like the hot. The effect of the equipotential bond is to dynamically change the bias of the whole transformer secondary, to keep neutral at earth potential and make hot move away from it.
Other useful reasons
A desired side-effect of the equipotential bond is that if there is a hot-earth fault, there is a high-current path via ground wire, conduit etc. back to the neutral-earth equipotential bond, and ultimately back to neutral. This completes the circuit, allows high current to flow, and causes a circuit breaker trip, which arrests the ground fault. Remember, current wants to return to source, not to ground. It doesn't care about ground, except that the equipotential bond makes it care.
For a variety of reasons, there needs to be exactly one equipotential bond. Another one would create redundant (paralleled) paths for normal neutral (return) current, and that causes all sorts of mischief.