Yes, that's a good way to visualize it. I don't think the video's representation violates that model though. But back to your original question "why light's speed is so special." That person on that very fast space ship going on a short trip; they would move slower through time (and faster through space) than everyone on Earth. An observer on Earth might watch them take over 100 years to make that short journey, but only a year might pass for the person on the ship. That difference between the two grows asymptotically as you approach the speed of light. General relatively says as you get closer and closer to c, and you had infinite fuel, that difference would grow to the point where the relative time changes across the length of the ship would grow so great that final drop of fuel needed to reach c would never make it into the fuel tank.