I don’t completely understand this video. It appears to me to be just the first of two necessary steps. In this first step the tire is made smaller more by buckling than upsetting.
It seems the second step should be to take another high forging heat at the buckle. The lock the tire in the shrinker with the buckle directly over the anvil plate and forge the buckle flat and smooth. The grip of the dogs prevents the forged segment from growing laterally, the hammering brings the tire back to proper dimensional thickness. The tire does grow some in width and that can be removed by filing after the tire has been placed back on the wheel.
I looked through some of my old photo files and couldn’t find an image to the machine once owned. A Google image search brings up a number of similar ones. I’m just beginning to recover from what I presume was the flu apparently not prevented by my flu shot, so, I have no energy to pursue it further at this time.