I've never heard a correlation between BMI and razor type, usually the discussions are about possible correlations between beard type and razor grind and size. But you're right that people with more rounded faces probably can get away with more pressure than guys with bony angular faces. It's probably also the case that they probably need to use more pressure as well to make their skin conform to the edge of the razor if they want to shave more than a 1/2" stripe at a time, especially if they're using a razor with a smiling edge.
I'm pretty sure I know the video you're talking about, it's the one with the mediterranean-looking guy shaving with the TI Pierre Theirs razor. He does seem to use a lot of pressure, though that's because his razor isn't terribly sharp (he mentions in the comments that it was due for a refresh at the time he made the video), and also because his heavy beard is just tough to shave. He's also pretty heavy so it wouldn't take much pressure to distort his face like that. I've got a heavy beard as well and my face distorts some (though not as much as that guy) even if I'm not using any pressure - even with something as sharp as a Feather. The particular razor this guy in the video is using usually has a smiling edge and he's got a pretty round face so he may use more pressure than you would just to shave a decent swathe of skin.