It seems to me that inertia is indeed the fundamental idea, and Young's modulus can be argued to arise fundamentally from the combination of inertia and sound speed (or the speed of whatever is carrying the internal forces). As an example, imagine a charged conductor in an electric field. The conductor will receive forces whereever there are charges, and the manner in which it compresses in response will depend on the details of the charge distribution, but the inertia of the conductor will not. Or in your truck analogy, imagine pushing at the back at the same time somone else is towing it from the front-- the compression will depend on how the forces are applied, but the inertia can be determined from just the total force alone.