I've heard this from several sources, though everyone seems to have differing opinions as to what point it stops helping. I know for deer up to 250 lbs I've seen fairly spectacular differences between a 243 at 3700 ft/s lung shot and a 45-70 with 2000 ft/s heart shot, but that isn't a heavy boned animal, I am positive I wouldn't see that same high velocity round instantly kill an elephant for example, but I don't think there is any definitive cutoff animal, everyone has their own opinion on that, and I would be interested to know what size (lbs) you think would be a cutoff, as you have obviously hunted Africa, while I have only hunted Kansas deer and coyotes, which are much smaller.
I will say I disagree with your evaluation of the video of a .308 doing the same as the 300WSM. I think moose is probably too large to be affected by hydrostatic shock, but not cavitation. Just to be clear cavitation is the direct movement of tissue due to energy and momentum of the bullet passing through the tissue. Hydrostatic shock occurs when cavitation size causes blood displacement greater than its ability to bleed out and sends a pressure spike though the vascular system. The smaller the animal the less surface area of veins in the body to stretch and disperse this pressure spike and the more likely it is to cause an aneurysm. Clearly the moose didn't experience an aneurysm, so hydrostatic shock didn't kill the moose. It is lung shot, as evident by the hot moist breath coming out both sides of the chest cavity, and yes I'm sure the .308 would have penetrated both sides just as well. However the cavitation on gelatinous lung tissue will be larger with the 300 WSM, and thus cause more damage to the lung, and faster death. How much faster we can debate, but there is no real way of testing as that animal is long dead. Now comparing the cavitation between a 338 and a 416 is much more difficult than two rounds of the same caliber. I would like to point out in your example, the 338 vs 416 with a 400 gn bullet, the 416 has much more energy and as such will have the potential for greater cavitation, so yes I agree I think a 416 on heavy PG should be much more effective. But then again so would a 600 NE, but I wouldn't advocate using it.
I agree with your analysis that a 338 Win mag is the best choice, I've seen lots of sources over the last few days to support that. I do believe under some circumstances a barrel burner will be a faster kill, but not considerably and not worth the cost difference so long as both rounds penetrate sufficiently.
Side note, I fully support your statement the hunter should have kept shooting. Some people are worried about ruining meat with followup shots but that's a small price to pay for a humane kill.