about the saying "neutral does not carry current"
That saying is wrong.  It's dead wrong.  The people saying it are idiots. Or you misunderstood. 

Imagine if you found an SE post that started out "People often say 'humans do not contain blood'. I think that's a false statement."  It's like that. 

Neutral certainly does carry current.  In a single-phase, 2-wire circuit with neutral, neutral carries exactly the same amount of current as the hot.   Neutral is the normal current return. 

Now in split-phase or 3-phase where 2+ hots and neutral are brought along, it's likely most of the load will be phase to phase, with neutral carrying only imbalance current.  In multi-wire branch circuits, all loads are phase-neutral and neutral acts as a broker to try to return current via another load rather than bringing it back to the panel itself. But neutral is still a normal and legitimate carrier of current, and it counts. 



A great deal of work is done to assure that Equipment Safety Ground is at roughly the same voltage as the dirt surrounding your house.  That way if you are touching a grounded receptacle while standing on your wet grass in your bare feet, nothing happens. 



We also make very strong effort to make neutral reasonably close to ground potential.  Nobody cares if neutral is 7.3 volts away from the dirt in your lawn. We just want it to be close enough to be safe. 

It's true if you are standing barefoot on your lawn and you touch neutral, you shouldn't be shocked. But that's not the main reason to do it.  The main reason is to assure that your "hot" wires are only 120V from the dirt in your lawn.  Consider the alternative.  Imagine some leakage in the transformer caused your supply to float 2400V above where it should be: Hot 1 would be 2160V from earth, neutral would be 2400V from earth and hot 2 would be 2640V from earth.  Now suddenly, touching any wire with your bare feet would be instantly lethal.  

By providing a firm neutral-ground bond, in one place, we assure that neutral will be near 0V from earth, and the hots won't be terribly far away either. 

Picking any supply wire and bonding it to earth in this way defines that wire as neutral.  Neutral need not be "in the middle".