It felt to me like the author was aiming at physics / comp-sci students who are already familiar with linear algebra, and may have less of an aptitude for working from first principles.

Though I have to admit not to know geometric algebra, from both you and the article, I gather it was created as an improvement on linear algebra. For those less innately interested in maths, starting from such a position tends to work much better than starting from scratch.

I feel like there are 2 methods aimed at different people. For the 'real' mathematicians, the approach you advocate is better suited. For those who use maths only as a tool, and do not so much appreciate the innate properties of maths, the easier to follow (but less informative) method of the author might be better.

Sadly, there is no single 'best method' for teaching a thing.