I don't understand the reason scales measure the normal force instead of the weight.
I'll give an answer based on intuition (which illustrates the physics).
The scale only measures the portion of the weight that is applied to the scale.
Imagine you tie a string to a weight and place the weight on the scale. If you pull up slightly on the string the scale will read less, even though the weight of the object has not changed. The weight is counteracted by both the normal force from the scale, and the tension you are applying with the string. But the scale only "sees" the normal force.
Scales measure the force being applied to their plate, through springs or circuits. The force on the plate of the scale is the force from the object on it. It is Newton's Third Law pair with the Normal force on the object.
The scale measures the normal force because the force on the object and the normal force are necessarily equal in magnitude, through Newton III.
The normal force is generally a proxy for mass, since it is often the sole force responsible for countering gravity when an object is at static equilibrium. During cases like an elevator where there is another vertical force, the normal force may be different from the weight force in magnitude.
When the normal force is different from weight, that's when it's important to note what a scale actually measures. The force the scale reads will always equal the normal force in magnitude, but not always the weight.
When a mass $m$ is placed on the ground and the gravitational field strength is $g$ the ground is compressed a little so that the upward force on the mass due to the compressed ground or the normal force on the mass, $X$ is equal in magnitude to the weight of the mass $mg$.
With up as positive, the equation of motion for the mass is $X-mg = 0$
If a moving mass hits the ground the upward force due to the ground being compressed or the normal force on the mass, $Y$ has to be larger than the magnitude of the weight of the mass so that there is a net upward force on the mass to make the mass accelerate.
The equation of motion for the mass is $Y-mg = ma$ where $a$ is the acceleration of the mass.
Scales are really no different from the ground in that inside a scale a spring is compressed and the amount of compression is related to the force the scale pan exerts on the object on top of it.
Using the ground directly as a scale is difficult because the deformation of the ground may not be elastic and might well be too small to measure.
Scales indicate the force on a mass on the scale due to the scale which is the same as the normal force on the mass.