I completely agree that weight is underutilized. It can be carried consistently within GA, and even if you only use its sign, you already gain extra descriptive power.
All uses you sketched should be possible and useful, and should be developed. For PGA, you find an initial attempt in the PGA4CS writeup. I am writing a piece about Newtonian mechanics by PGA (inspired of course by Gunn) that uses the weight as mass. And indeed, relative importance of data in linear operators is naturally done by weighting them.
In PGA, the neglect of the sign is a sin; it happened in projective geometry by homogeneous coordinates as well, even though it can tell you whether a point ends up in front or behind the camera. We need the signs, and would be hacking them in. Let us treat them structurally.
Leo