Well, a rotation matrix doesn't require doing 2 half rotations, and doesn't require reaching into the 4th dimension in such a way that it gets perfectly cancelled out. It doesn't require abstract analogies about cubes with strings glued to them or coffee cups.

With some familiarity with linear algebra, it's easy to derive the formula for constructing a rotation matrix. You just have to think about what the operation does to the axes. The derivation for quaternion rotation is far more abstract, by virtue of the operation we actually care about involving a sandwich of multiplications with unclear 4 dimensional meaning.

Augmenting your space to handle not just rotations & scaling, but translations is easy for matrices, just requires a homogeneous coordinate and you get 4x4 matrices with intuitive columns.

Augmenting quaternions to handle translations requires the 8 dimensional dual-quaternions.