Hartman-Grobman isn't so much about approximation error by choosing different simplifying models. It's about being able to do some analysis on the linear approximation with the results carrying over to the original non-linear system. In particular, it lets you categorize stationary points of a non-linear dynamical system, using only the linear approximation at the fixed points, which is awesome since analyzing linear systems is super easy.

My take on this theorem is that it is most interesting to know when the topological equivalence doesn't hold, i.e. linear approx of equilibrium point can be poor when there is the possibility of a limit cycle oscillation.