By identifying Hamiltonian flows with geodesic flows of suitably chosen Riemannian manifolds, it is possible to explain the origin of chaos in classical Newtonian dynamics and to quantify its strength. There are several possibilities to geometrize Newtonian dynamics under the action of conservative potentials and the hitherto investigated ones provide consistent results. However, it has been recently argued that endowing configuration space with the Jacobi metric is inappropriate to consistently describe the stability/instability properties of Newtonian dynamics because of the nonaffine parametrization of the arc-length with physical time. On the contrary, in the present paper, it is shown that there is no such inconsistency and that the observed instabilities in the case of integrable systems using the Jacobi metric are artifacts.