I don't think you're missing anything. One of the goals of this article is to demonstrate how geometric algebra is really just another part of exterior algebra anyway. However, performing transformations in spacetime with the geometric product might make some things easier to understand, but I'm not making any strong claims about that. This article contains a small glimpse of what I'm talking about: https://terathon.com/blog/relativistic-quaternions.html

Since Geometric Algebra == Clifford Algebra, Michael Atiyah, Roger Penrose, and Paul Dirac disagree with you.