[fault1]

You certainly do not need to have a phd in geometry and algebra for this. For example, the exp map is used all the time in robotics particularly in rigid body dynamics and kinematics.

Here is a free book made for undergraduates that teaches in that manner that just assumes some elementary linear algebra; http://hades.mech.northwestern.edu/index.php/Modern_Robotics

IMHO, geometric algebra makes certain things clear, but it's also oversold as something new or novel. It can be recast in the language of differential forms (covariant multivectors) which is very often used in physics.

[ajkjk]

For what it's worth, differential forms have plenty of their own pedagogical problems. IMO most of the value in them comes from the concept of multivectors, and very little from the fact that they're 'dual' to regular vectors (which is a finicky detail necessary to do covariant geometry correctly, but not useful for general intuition).