Sure they can fit any experimental result that way, they can probably fit any 19 experimental results that way, but in general if you would freely adjust a constant to fit one experiment then it would stop fitting other experiments.
Yes. This would be extremely helpful for experimentalists who spend a lot of their time pointing out, for example, that one's new theory can't violate the equivalence principle by very much at all. Similarly, it would be helpful for people planning new experiments to know whether or not their proposed experiment will probe new ground (i.e. CERN's anti-hydrogen experiments are of intense value for spectroscopic studies, but existing experiments [1] show that antimatter, at the 10^-8 level or better, obeys the equivalence principle and therefore will reliably fall in every experiment of which CERN is capable.).
As a sibling comment points out, it is difficult to implement the markup that spans the space of all possible theories. Kostelecky's parametric Standard Model Extension offers one avenue to do so.
One could implement such a test as a checklist, too, which might already make a difference.
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 :) )
