You're right that a nearer small body pulls the large body with a greater force, but after one full orbit the large body has been pulled rightward just as much

You're right that a nearer small body pulls the large body with a greater force, but after one full orbit the large body has been pulled rightward just as much as it has been pulled leftward. The missing piece is: for how long is the force exerted before the situation repeats? A distant small body exerts a smaller force for a longer time, and a nearby one exerts a greater force for a shorter time. It comes down to how much time and how much force. The law of graviton plus Kepler's third law plus some algebra ought to show that you get a larger circle with a weaker force over a longer time (not an obvious result).