The transfer matrix of an optical system, by definition, is the Jacobi matrix at a single point in optical phase space of the general transformation wrought by the system. It therefore describes the system's behavior linearized about that point. In other words, it holds only for rays that travel paths through the system that are near to that of the "chief ray", i.e. the ray that defines the linearization point.
Thus the transfer matrix can only "see" linearized behavior. All lenses described by a $2\times 2$ transfer matrix are therefore perfect: rays converge perfectly to their focusses when described by the transfer matrix. A $4\times 4$ transfer matrix, tracing the $(x,\,y)$ positions and $(\theta_x,\,\theta_y)$ angles of rays, can account for an astigmatic lens: different focusses for the meridional and sagittal planes. By definition, it cannot "see" higher order aberrations, so astigmatism is the only aberration that be modelled.
Even tilts are awkward to model, because a tilt is an inhomogeneous transformation - a constant angle added to ray angles and matrices model homogeneous linear transformations. You can model a tilt by adding the offset to the transformation and then taking it away again. If you do this for a thin lens, the procedure has no effect, reflecting the approximation that a ray through a thin lens's optical center is undeviated. If you do this procedure with a thick lens, i.e. one with a nonzero distance between its principal planes, it will correctly model the deviation caused by the rotation of the "lever arm" between the two principal planes.
Aberrations must be calculated either by full ray tracing or, as you have seen, various ad-hoc correction terms and procedures added to the transfer matrix method. Transfer matrices are most useful in calculating imaging geometry and magnification. They are also symplectic matrices, which means that they also model the important principle of conservation of the optical (Helmholtz) invariant and conservation of étendue and the limitations imposed by these principles on imaging geometry and magnification.