Caveat emptor: he’s doing a radio show, so there’s a necessary downconversion of the depth of the discussion. He does err when he implies that massive objects traveling at c is a technological/engineering barrier, and we just haven’t figured it out (like breaking the sound barrier), rather than a physics barrier that would demand that current theory be wrong or incomplete, with some new physics necessary to explain what’s going on.
In general the “photons don’t experience time” is a bit of a tiptoe in the mine field, because we can’t definitively say what things look like from a photon’s perspective — relativity doesn’t afford us a description of what things look like to a photon. The Lorentz transforms — what we use to see what things are like for massive objects — “blow up” when v=c, which means that the photon is not in an inertial frame, and all that physics we do is for objects in inertial frames. The reason I hesitate is that an equation diverging from some infinity showing up means that the equation has failed, so the fact that a term is tending toward infinity (or zero) as it fails isn’t a guarantee that infinity (or zero) is the right answer when the equation no longer applies.
The idea does paint a nice picture, though, and seems consistent with the QED path-integral idea that photons sample all paths and most of them destructively interfere, and what we see is the result, but I think it’s easy to take the idea too far and imply things that don’t have any experimental backing to assure us they are true.