The reason it is not taught is because geometric algebra (Clifford algebra to the rest of the world) is a specialized tool for producing certain representations of the rotation/Lorentz group, and it does not have a distinguished place as a defining algebra of space-time.
What Hestenes does is reformulate everything using Clifford algebras instead of the usual coordinates. This is like taking the "slash" of every vector. You can do this, but it is difficult to motivate. Hestenes' motivation is to make an algebra out of the space-time coordinates. But it is not at all clear that one should be able to multiply two vectors and get something sensible out. Why should it be so physically? The reason is clear when you have Dirac matrices, but to introduce this as an axiom is unmotivated from a physical point of view, and I do not think will help pedagogically.
Further, this formalism, while it naturally incorporates forms, has a hard time with symmetric tensors, which are just as natural as antisymmetric ones. You can represent symmetric tensors, of course, using spin indices, but there is no reason to prefer the Clifford algebra way, because you can use other formalisms with equivalent content.
To produce an algebra out of vectors and to claim that it is physical, requires an argument that vectors should multiply together to make antisymmetric tensors plus a scalar. This argument is lacking--- the scalar product and the wedge product are two separate ideas which do not need to be combined into a Clifford algebra unless you are motivated by spin-1/2.
I think that this idea is cute, but it is not pedagogically useful in itself. It might be useful as a way of motivating the Ramond construction in string theory (this is just a hunch, I don't know how to do this), or other otherwise mysterious Dirac matrix intuitions.