My understanding is that with the lagrangian approach then the free parameters are not all interacting with each other because they are part of different terms. This means a change to a free parameter doesn't necessarily break experiments.
The point is that there are now 10s-100s of experiments that have been reported to very good precision (obviously not all to the extra-ordinary precision of this measurement). There are no longer any “free parameters” in the SM, in the sense that each one has been constrained by at least one experiment by now. Also, in complicated processes like this one, multiple parameters could make an effect on the observed value, such as the fermion masses. (Not saying the fermion masses actually affect g-2, it’s been a few years since I’ve done any QED, so my memory is a little cloudy :) )
