I think the exponential map is still the most robust way to think about rotations, since it gives you tools to deal with the Lie group of SO(3) in the most straightforward way (switching between coordinates, dealing with differentiation and tangent spaces, etc.)
Even when going through all the formulations with geometric algebra, you still land on using rotors (isomorphic to quaternions) and motors (isomorphic to dual quaternions) to represent SO(3) / SE(3) spaces - but I think for that purpose 3x3 rotation matrices / 4x4 transformation matrices with exponential maps are still much more useful. (Quaternions do have an advantage that it stores up less space and also faster to multiply with each other - but when you want to transform points with it then matrices are still faster. In overall in terms of efficiency it really depends on the situation.)
Even when going through all the formulations with geometric algebra, you still land on using rotors (isomorphic to quaternions) and motors (isomorphic to dual quaternions) to represent SO(3) / SE(3) spaces - but I think for that purpose 3x3 rotation matrices / 4x4 transformation matrices with exponential maps are still much more useful. (Quaternions do have an advantage that it stores up less space and also faster to multiply with each other - but when you want to transform points with it then matrices are still faster. In overall in terms of efficiency it really depends on the situation.)