> As spekcular says, the standard curriculum covers differential forms, tensors and vectors... making geometric algebra a relatively small delta to pick up.
Could you be a bit more specific about which "standard curriculum"/"standard approach" you're talking about?
For example, in my formal education (high school; masters with physics major, comp. sci. minor; 4 years of a comp. sci. PhD (abandoned)), I did not encounter differential forms, tensors, multivectors, the wedge product or multilinear algebra (or quaternions, lie derivatives, differential geometry, (co)homology, etc.).
Maybe you're talking about a "standard approach" for a pure mathematics curriculum, or perhaps physics/math grad school?
All I can say is that high school and undergraduate physics (in the UK, circa the late naughties) (a) does not standardise on those topics, (b) is filled with tricky operations which are easy to mix up or perform the wrong way around (e.g. cross products, matrix multiplication, pseudovectors), and (c) many of those annoyances would simplify-away under GA.
It's a cliche that physicists (certainly when teaching) cherry-pick the parts of mathematics they find useful. All of those concepts would certainly be useful in a physics course, but would perhaps be too much to fit in; yet there's certainly enough scope to cherry-pick GA (since we can drop Gibbs-style vector algebra[0] to make room). Perhaps something else, like differential forms, might be even better; I honestly don't know (maybe I'll do some reading about it).
[0] By "Gibbs-style" I mean the 'cross product and dot product ought to be enough for anyone' approach that permeated my undergraduate learning.