>As I understand it, this is sort of simulating what it would be like to capture this, by recreating the laser pulse and capturing different phases of it each time, then assembling them; so what is represented in the final composite is not a single pulse of the laser beam.
It is not different phases, but it is a composite! On his second channel he describes the process[0]. Basically, it's a photomultiplier tube (PMT) attached to a precise motion control rig and a 2B sample/second oscilloscope. So he ends up capturing the actual signal from the PMT over that timespan at a resolution of 2B samples/s, and then repeating the experiment for the next pixel over. Then after some DSP and mosaicing, you get the video.
>It seems like if you could measure the pulse's propagation in one direction, and the other (as measured by when it scatters of the smoke at various positions in both directions), this seems like it would get around it?
The point here isn't to measure the speed of light, and my general response when someone asks "can I get around physics with this trick" by answer is no. But I'd be lying if I said I totally understood your question.
[0] https://www.youtube.com/watch?v=-KOFbvW2A-o