Hmm… I still don’t understand how slapping a flat tire on a rim changes the distance traveled. Rolling a bare rim next to a rim with a flat tire on it, the bare rim will spin faster?I'll let you do that experiment! But if the tire isn't slipping on the rim or on the ground, then ya... close to the tread length. I think you'll need to put some air in the tire though, to satisfy those criteria.
Yes the test was crude. It would have been better to deflate it more. But he showed that the difference in travel/rev was very tiny... much less than the radius reduction.
The "factor" you are overlooking is that the rolling circumference change with pressure is actually a tiny modification of "0".My conclusion is that a flatter tire definitely changes the “rolling radius,” therefore, distance travelled. But perhaps not a 1:1 ratio I was thinking of. Probably some other factors involved I’m overlooking.