The speed of light traveled is all relative to time outside the speed of light. It’s continuous. Time inside and outside the speed of light is the same. If light belongs to the same space/time continuum then there is no difference.
Time slowing down is just a theory. It cannot be proven or disproven at this time. Yea the physics add up but common sense says that it cannot be true; for if it was true Space within this continuum would be ripped apart by time. Even in a black hole this is true, yet, I think its gamma radiation that escapes violently from black holes.
Look a human being cannot survive the speed of light nor the ripping of black holes. If we look from an outside perspective we can see that it all happens within time. The passage of time is a constant it doesn’t change no matter what physical reality works upon it. Perhaps, the perception of time passing plays tricks.
Let’s say we have a photon that has to go hundred meters. When the photon hits the hundred meter mark it has done so through the same passage of time whether within the speed of light or without the speed of light. We can perceive this to be true in either perspective; otherwise it becomes paradoxical because even though the photon hit the hundred meter mark, in your theory it would still be travelling to hit the mark because time slows. Time relates to time inside and outside of the speed of light. its all the same.