I always love talking about this subject. Biomechanics is a fascinating subject in and of itself. First of all consider your first sentence . The Dog is actually PUSHING the harness which transfers the momentum to the cart through the traces and/or the shafts depending on how you are set up. At no time does the dog pull the cart strictly speaking. As for what does the work, it take both the fore and aft legs to get forward motion. Looking at the physics of it, the rear legs contribute more towards the forward motion because of more muscle mass and the leverage of the muscles that contribute energy onto the harness. Because of the angles involved, the front legs would never be able to generate the same forces as the rear legs because their position relative to the front of the harness. Just think of it as a jumping exercise in which the feet remain of the ground. In a jump the front feet do lend initial momentum to get the body into proper attitude for the jump but the rear legs do the majority of the work to get up and over. Just consider how you feel when you try to lift something with your back instead of squatting over to lift with the legs. You are more powerful using your legs and could put your back out, doing otherwise. A good resource for this is "Jumping from A to Z", by Christine Zink. Just think jumping on the level ground!
Jim LaFrom San Jose, Ca now get out there and work dem workn'dogs