In the (modified) MS renormalization scheme, after dimensional regularization, we introduce some parameter $\mu$ with power of mass to keep the dimensionality of integrals under control. The parameters in the lagrangian end up being functions of $\mu$, and from the beta functions we can compute how they change when we change the "scale" $\mu$.
Now that's the point: we say that perturbation theory "breaks down" at high energy in QED because, if we identify $\mu^2 = s$, we end up with an effective fine structure constant which is of order one, hence of course we can no longer make use of perturbation theory.
If that's the case, why do we choose such a value of $\mu$? If it's a fake parameter that we put in our theory to "fix" an issue we have with dimensional regularization, why don't we pick a different value of $\mu$ which is very different from $s$, rendering $\alpha(\mu)$ very small?