I dislike the term "rideing the force of the push," as it is misleading. You can't ride the force forward, as you point out. However, you can void your left side, leaving him with nothing to push against. This allows you to step forward not so much under, but around the force of the push.
TS is also a lesson in pivoting a strike at the trailing shoulder instead of around your centerline, which is much more alive. It lends itself to moving off center of his attack, gives more penetration of his space and enhances power generation. Carry that concept through the elbow and back knuckle strikes and you'll notice a huge difference.