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OK, so in the second part of your post you have finally acknowledged that weight is a measure of the FORCE the earth's gravity exerts upon a mass. Walking you gently through this, the scale is measuring that FORCE.
As you correctly point out, weight is referring to one specific force, gravity (or at least its effect on a mass), which is why newtons are used for measuring force in other scenarios. As the gravitational force on a 1kg mass on Earth is 9.8 newtons, you could attach your kitchen scale to the wall, press on it with your hand, and when it reads 1kg you're exerting approximately 10 newtons of force on it. 2 kg = roughly 20 newtons. This is how a scale can be used to measure force even though it reads in KG and not newtons. All a force meter is is a scale calibrated to display in newtons.
The amount of force required to push a blade through a consistent medium can be taken as a measure of its sharpness as a sharper blade will cut through more easily, hence if a blade requires 1 newton of force to cut through 1cm of medium, and another blade requires 5 newtons of force to make the same cut, the conclusion can be drawn that the blade which requires less force is the sharper of the two. This is how force can be used to compare sharpness, and how a scale calibrated in KG can be used to measure force.
I've done my best to clarify this, but if you're still not following I don't think I can help.
Newton has also done his best to clarify this, but he is dead and i am tired. What you don't understand, is that when a blade cuts, it isn't subjected only to earth's standard acceleration. It has it's own acceleration the moment it cuts through. Which cannot be measured in weight (kg), because in weight, the only acceleration is one, the standard g (9,8 m/sec) of the earth.
This is why force is measured in Newtons and not in kg. Weight is measured to an object only when it is subject to the sole acceleration of the earth's gravity. On anything else, you measure force, in Newtons. And this is why factories pay for special machinery to measure sharpness, instead of rigging a scale and do that for pennies. And this is why when you hope on body scale, you don't move around to get the reading. A blade while cutting, is accelerating (see "peak force" in the link i posted above. If mass remains the same, how do you explain the existence of "peak force" in the type F=mass x acceleration? Surely, the blade isn't changing mass, right? Then what's changing? The acceleration of the blade. And this is why they use the paper medium in all the "normal" sharpening measuring devices and not a scale or static device.
On a sidenote, this is from another guy, apparently conducted in a lab with a self-invented mechanism. Strange that shows Feather and Kai as applying the max force, huh, while the guy with the scale has Kai close to Derby, isn't it...
https://mielabs.coe.neu.edu/sites/mielabs.coe.neu.edu/files/Razor Blade Sharpness Testing.pdf
Like you said, the scale, measures weight, which measures the force of earth's gravity, under the influence of earth's accelleration, which is fixed. That's weight. You don't measure sharpness by weight or cutting force for what matters.
Either that or the people who don't use scales are morons and pay money for useless equipment. And blades get sharper after shave #3.
That's all from me...
EDIT: Useful reading:
https://imada.com/what-is-a-force-gauge/
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