And besides, the question is over the entire proof strategy and not the specific techniques involved. It seems plausible that one could give a relativizing proof using some method of calculation from statistical mechanics, for example.
Such proofs are said to relativize -- i.e., they are still valid relative to any oracle.
Taking a methods from a different domain in no way shows that the mechanics of those method don't reduce to the same mechanics as a "natural proof". Further, it makes the actual process much more obscure.
Indeed, statistics in general seems like a hard approach for overcoming the Razborov-Rudich limit, since the limit is on showing that a "typical" function has certain properties and anything statical seems like it would be "typical".
But who knows really, maybe Razborov-Rudich isn't correct, since in fully generality it relies on things that "mathematicians generally believe" (the existence of certain pseudo-random functions) rather than things that are definitely proven - as far as I know.
"Model theory" might be closer to "a" technique or at least a somewhat distinct set of techniques.
And other have noted, there's reason that a proof from statistical mechanics would relativize.