I think Feynman always overlooked (edit: or "underplayed", perhaps) the fact that he was - well - Richard Feynman. He had pictures in his head that worked right because he understood the maths to the point that he didn't need to do it explicitly anymore. If you don't understand the maths, your pictures probably aren't pictures of the real world.I know Feynman was an accomplished mathematician, but he seemed to think in visualizations. Geometry and diagrams first, math proofs later.
The problem is that you are imagining an inertial frame of reference that moves at the speed of light. However, the speed of light is constant in all inertial frames of reference. That means that in a frame moving at the speed of light, light is travelling at zero and 3x108m/s at the same time - which is nonsensical. The conclusion is that there is no such thing as an inertial reference frame moving at the speed of light. The infinities that pop up when you try to figure out what's going on in an inertial frame moving at the speed of light is the maths' way of telling you that you are doing something illegitimate.Will this help you help me understand my error? Thanks again.
Given that you can't have an inertial reference frame travelling at the speed of light, you can't really "reason your way through" it to what happens (or not) at hypothetical speeds greater than c. I think you'd need to develop a theory that does permit self-consistent descriptions of travel at the speed of light then show that it reduces to relativity somehow before you could reason like that.