Would knowing the exact weight that has been loaded on a commercial plane and it's distribution be useful?
YES.

The obvious solution is to roll the plane over a scale.  Useful CG data can also be obtained by weighing the front wheel and comparing its torque to known values.  A light front wheel means an aft CG.  This could be corrected by moving fuel BEFORE the plane is airborne, as well as while in flight.

Of course, moving passengers and cargo are an option too.

For example the plane weighs in at 2200 lbs total on the ramp ready to go.  The nose wheel weight is 200 lbs.  The distance between the main and nose wheels is 11 feet.  The aft CG limit is 0.75 feet forward of the mains.

CG is calculated as follows:  200 lbs/2200 lbs = 1/11. 
          CG is 1/11 * 11 feet = 1.00 feet forward of mains. 
          CG is with in limits.

And how accurate weighing the nose wheel can be (due to long torque arm):

Nose wheel weight 100 lbs.  Total weight 2200 lbs.
        100 lbs/2200 lbs = 1/22.
         CG is 1/22 * 11 feet = 0.500 feet forward of mains.
         CG is NOT in limits.