Why a 4x4 homogeneous transform instead of separate R and t?

In robotics we represent a rigid-body pose as a $4\times4$ homogeneous transformation matrix $T = \begin{bmatrix} R & t \\ 0\,0\,0 & 1 \end{bmatrix}$, where $R \in SO(3)$ is a rotation and $t \in \mathbb{R}^3$ a translation.

Explain the concrete advantages of this homogeneous representation over storing a rotation matrix $R$ and a translation vector $t$ as two separate objects. In your answer, address: (a) how composing transforms across a kinematic chain works and why associativity/order matter; (b) how you invert a $T$ and why the closed form is cheaper than a general $4\times4$ inverse; and (c) at least one practical pitfall of the matrix form (e.g. numerical drift off $SO(3)$, memory/redundancy, or singularity behavior) and how you would mitigate it.