Couple of points he's got wrong - camera-subject distance doesn't matter for camera shake discussions. Until the light rays hit the lens they're traveling in a straight line, and aren't exposed to a "shaking" medium (ignoring the effects of thermal refraction). It's only when they hit the lens that there's any shake issue to deal with, because the camera is what's moving, not the outside air. Actually the light rays are still traveling in a straight line (subject to refraction in the lens); camera shake causes the sensor to move over the duration of the exposure. All any anti-shake mechanisms do is try to deflect, either physically or logically, the incident light rays to counter the camera motion.
The argument's being made that since you use a longer lens on FF, there will be more shake. Yes, but, this is a red herring. The actual linear distance may be greater, but the angular motion tends to be pretty consistent across focal lengths. Your shake may be 0.01 deg. of arc over a 1/200 sec. exposure. If you map 0.01 deg. of arc to the actual "smear" of your point on the sensor, it'll cover 0.035 mm over 200 mm. If you map 0.01 deg. of arc from the 300 mm lens to the actual "smear", it's a blur of about 0.05 mm. When you enlarge these two to get an 8x10 print, you enlarge the "negative" 13.5x for an APS-C camera, but only 8.47x on the FF camera. That gives a print "smear" from a 200 mm lens on an APS-C sensor of 0.47 mm; on a FF with 300 mm, it's 0.42 mm. And a "point" that's about 0.5 mm is blurred enough to be "unsharp". So you really do need to figure in the "crop" in calculating hand-holdable speed. For the same reason, "crop" (or final print size) has to be considered in determining depth of field. The "circle of confusion" varies with the sensor format. The more you enlarge a print, the smaller the acceptable circle of confusion in the original negative, subject to viewing distance (which tends to be consistent fro any given size print).