You might break science.
We already use fundamental constants to define other important measures. The article gives the example of a meter, but a second has also been standardized this way. A second is defined using the natural decay of a cesium atom. These measures mean that anyone, anywhere, at any time can verify a measurement with the proper equipment. The problem with gravity (and hence measures of weight) is that gravity is incredibly weak. Think about it. If I throw a rock, I can temporarily defeat the total gravity of the entire mass of the planet earth. There are so many things that can counter gravitational force to one extent or another that the precise work needed to obtain a universal constant is not easy.
I also can't help but thinking that there must be a weight loss joke in this story somewhere...
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