A bias towards 'vectors must be points' crops in as an unwritten legacy axiom, leading to a broken correspondence with Group theory (which again cannot be turned around, discrete symmetries compose into continuous ones, but not the other way around). A further restriction to the unfortunately very symmetric 3D Euclidean space hides many of the problems with that approach. (for example only in this space the even subalgebra is shared, and bivectors are self-dual).

Consider for example R_(2,0,1) in both scenarios.

Using Geometric Product as Group Composition

  e1  = reflection w.r.t. x=0 axis
  e2  = reflection w.r.t. y=0 axis
  e12 = reflection w.r.t. y=0 axis followed by reflection w.r.t. x=0 axis
      = 180 rotation around origin.

Using Anti-Geometric Product as Group Composition

  e1  = not a group element, because anti-norm is zero.
  e2  = not a group element, because anti-norm is zero.
  e12 = not a group element, because anti-norm is zero.

So to get the same behavior you need the following elements when using the Anti-Geometric product as composition

  e02 = reflection w.r.t. x=0 axis
  e01 = reflection w.r.t. y=0 axis
  e0  = reflection w.r.t. y=0 axis followed by reflection w.r.t. x=0 axis
      = 180 rotation around origin.

So choosing the geometric anti-product as group composition operator is possible, but imho breaks the readability of your transformations completely.

All I can say is that I hope that those linked articles do not give people an excuse to not consider the model in which vectors, grade-1 elements, are reflections. This is the true half of the picture that is new and being missed.