Soma would have been called Tom Cruz. 
POST OF THE FUCKING YEAR.
On a joking note, I'd imagine that it would be like any American-based monster movie. Troops get pumped in like hosewater, and those in power can't quite grasp that they can't actually just shoot it to death, and then at the last minute they let the one guy take care of everything.
On a more serious note, I think it would hurt the romanticized atmosphere to the game. Not even going into the wardrobe department (and on that note, X, I think you're being a little bit too pessimistic on this; because I've seen men wear far more extravagant things than a fur-collared coat and they didn't "get the shit beat out of them." That's a really uncool mindset, dude. Just sayin'.
I would wear that coat, and would rock that look.), a lot of the themes going on in Aria would either feel forced or entirely nonsensical. For instance, the Hakuba Shrine and its (admittedly small) role. The US doesn't have shrines like Japan does (ours have price tags and food courts), so Mina's entire plot role would lose a lot, if not all, of its sensibility.