^I don't think you can treat the motion of a pendulum that asymmetrically. However high you manage to get on one end, you're going to get nearly as high on the other end, minus whatever momentum you lose to friction, deformation, and other inefficiencies. So it doesn't matter which direction you exert more force in.
When it comes to pendular motion, what matters is how high
you get above the resting point, not how far to the side. Because what actually imparts the acceleration is gravity. Remember equal and opposite reactions. As you saw, when Adam and Jamie tried to move the cage on their own, all they did was rotate it around their mutual center of mass. You can't actually push yourself sideways on a swing without touching the ground, any more than you can pull a car forward by dangling a magnet in front of it. But if you can raise the center of mass and move it
off to one side, then gravity will pull it back down and the pendulum's string will impart a centripetal acceleration that will convert some of that downward pull into lateral motion; and then you'll overshoot the starting point and the centripetal pull of the string will curve you upward, so that gravity will pull you down again, and so on.
So when you're on a swing and you stick your legs out on the upswing, that's not about thrusting laterally, it's about raising your center of mass higher. The higher your CoM is at the top of your swing, then the more acceleration you get from gravity on the way down. If you bend your knees back under you on the backswing, that has the same effect of raising your center of mass. The same if you pull back on the chain on the forward swing or push forward on it on the backswing, so that you tilt your body even higher.