To me, the horse's role is merely to transmit the force it is being pushed forward with by the ground over to the cart. So, let's say, the ground pushes horse with 20 N, the horse pulls cart with 20 N, the cart pulls horse with 20 N. If the force that the ground pushes the horse with is doubled, the force on the horse by the cart and on the cart by the horse should also be doubled and we are still left with 0 net force.
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If the horse pushes the ground with a force of 20 N, then the ground also pushes the horse. Hence, the horse would have a forward motion. In addition, if you draw the Free Body Diagram of the cart, you will find that the net force acting on the cart is in the forward direction.
Zwolf
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