I had originally thought so as well, but the people on the Paizo forum said that taking Firearms Training isn't necessary to qualify for the Grit Talent in this particular instance (hence why I didn't have it listed). I could try posing the question again on their forum but I'm not sure if I'll get more than a single reply.
I can't remember why I had Quick Draw though...this was a Weapon Cords build after all...