Pitching Moments CG vs CP vs AC
The root cause is the nature and theory of aerofoils. The design mission is to use the wing in an optimal way and finally the wing + body combination in the most optimal way to achieve the design goals. Most of the design goals answer to the following questions:

do you want to go fast (speed), or
do you want to lift more weight (payload),

For a given amount of fuel

do you want to go as far as possible (range), or
do you want to stay in the air for the longest period (endurance), OR, combinations such as
do you want to carry a specific weight a specific distance?

There will be an optimal aerofoil for each of these, and new designs can be attempted if there isn't one that fits the requirement or it is felt that there's room for improvement.
There are some aerofoil properties that tend to be common for the bulk of the commercial jet designs, as most of them are chasing similar goals:

generally best efficiency (in terms of L/D) is around 4˚ to 6˚ AoA. This angle is  achieved by combining a level flight pitch attitude of 2˚ to 3˚ with the wing mounted at a 2˚ to 3˚ Angle of Incidence**.

Stalling AoA in clean configuration somewhere around 15˚+, and

most of these aerofoils even produce lift till 1 or 2 degrees below 0˚ AoA.

With high lift devices extended, the typical AoA could vary between 7˚ and 10˚depending on slat/flap configuration.


(** Angle of Incidence: wings are mounted with the leading edge higher than the trailing edge, so the angle between the average chord line and the longitudinal axis of the airplane is fixed and it is called the Angle of Incidence. Thus an airplane rolling down a runway already presents an AoA of a few degrees and inflight, we don't have to cruise at an uncomfortable pitch attitudes of 5 to 6˚ nose up )
The physics of aerofoils and aerodynamics, as mentioned above, dictate that, the airplanes are mostly flown at different nose high attitudes of upto 20˚ including the slat/flap extended regime of flight, and climb phase. Much lesser part of the flight is spent at negative pitch attitudes of 0 to -4˚ for descents. Thus the 'lopsided' stabilizer range.
The Stabiliser settings for take-off depend on the CG, GW (Gross Weight) and T/O Thrust for the flight so as to present a 'standard' stick force and control effectiveness for the pilot and it must be understood that though weight acts through the CG, the GW is actually acknowledging the Lift being created by the wing to lift that weight.
BTW, a symmetric aerofoil produces no pitching moment only at zero Angle of Attack. Mount it at a small angle of incidence and it will produce a lift component acting through CP. It's just that there are other aerofoil shapes that have fewer compromises affecting the typical design goals.