The shininess would probably change the emissivity, so the camera probably can't give an accurate surface temperature for both sheets at the same time. That difference in emissivity probably does impact the heat radiating into the food, but I would guess the conductivity isn't really affect.

If I'm wrong though I'd be interested to hear what's actually causing the huge difference in the thermal camera shot. Maybe they just put the new one in for less time to exaggerate the effect?

I'm fairly skeptic about the physics of this demonstration. I'm a third year PhD student specializing in optical devices (I do a lot of work with infrared, which is what the thermal camera he used detects). I believe the 350˚ reading he got from the shiny pan is likely in error for two primary reasons.
First, shiny metals are not very good black body emitters (which the video indirectly acknowledged when he talked about how the pan absorbs less thermal radiation). This leads to slow and inaccurate readings in the camera's microbolometers (which are the devices used for measurement by all cheap thermal cameras). When the thermal coupling between the microbolometers and the pan is low, thermal conduction from the bolometer can lead to inaccurately low temperature readings.
Second, reflection off the pan can lead to radiative thermal coupling between the bolometers and the low-temperature background of surrounding room. Thermal radiation is emitted by the room's walls, bounces off the shiny pan, enters the camera, and reduces the pan's apparent temperature as measured by the bolometer. You can actually see the background reflection in the video as he moves the shiny pan. I ran into this issue a few months ago in my research and had to calibrate measurements to account for background interference.
So, as long as he left the pans in the oven for a fairly long time, they are almost definitely at the same temperature, and the measured temperature difference is an error. If he wanted to test this, he could leave the pans in the oven for an extremely long time (at least a few hours) to ensure they reached thermal equilibrium and were almost the exact same temperature. Then, if he did the same measurement with the same result, it would confirm that my objection is correct. Now, I also admit there's a chance I am wrong in my explanation, and the only way to be sure is additional testing. But it is certainly not true that shiny pans can't get as hot as dull ones.

I believe the video is correct about the superiority of the dull pan for cooking. The dullness increases its thermal radiation, which increases heat transfer from the pan to the bread, so a dull pan will likely perform better even if both pans are at an identical temperature.