We consider ergodic McKean stochastic differential equations with a unique stationary state and study the linearized (in the sense of McKean) diffusion process obtained by replacing the law of the nonlinear process with its unique invariant measure. We prove that the law of the nonlinear McKean process and its linearized counterpart are exponentially close in time, both in relative entropy and in Wasserstein distance. The analysis, based on entropy estimates and logarithmic Sobolev inequalities, is carried out on both the whole space and the torus. We then show how the resulting linearized diffusion can be used to replace the original nonlinear process for tasks depending on the long-time behavior of the dynamics, with a particular focus on parameter estimation from a single observed long trajectory.