For Seidel aberrations of thin radial gradient-index (GRIN) lenses, approximate expressions that have the same structure as the homogeneous thin-lens expressions but include also specific gradient effects depending on lens thickness have recently been obtained. In this paper it is shown that the approximate thin-lens formulas provide new insight into the possibilities and limitations of aberration correction with radial gradients. When lens parameters are varied, the change of the exact values of the Seidel coefficients is remarkably well described by the approximate expressions. As a simple example, aplanatic correction with a single radial GRIN lens is investigated. Features of several solutions that have been optimized with ray-tracing are explained with the thin lens approximation and exact Seidel correction.