It depends on where you are going to shoot the animal. I've seen 30-06 bullets not pass through a deer when crushing through bones on a double shoulder shot. My .300 has always passed through and dropped them dead. Dead is dead, but from what I've seen at my camp among friends it's a matter of how far do you want to track an animal
At the ranges listed by the OP there is no difference. Past that its pretty small improvement for a much higher cost in cartridges, rifle and barrel life (if you target shoot, hunting a sight in will not be enough to wear the barrel out). Hand loaded 30-06 with the right progressive powders will match factory loads.
The difference is the bullet and how it performs. You have to have the right bullet for the game and distance you are shooting at.
In the case of the above, he is obliviously using the wrong bullets in the 300 for his shooting and the 30-06 did what it was supposed to (expended all its energy on the target) . If the 30-06 had been a good heart lung shot maybe not. It was a kill shot, just not a good one.
The 300 would likely have done the same thing if hit in the same spot angle etc (it may have gone through but it would not have killed quickly)
There is nothing magic about velocity. Shots going through are wasted effect. Poking holes through things eventually causes them to bleed out but thats not the goal. Broken down shoulders stops game as well. You may have to do a mercy shot though the game will not have gone anywhere unlike a shot through wound if it misses heart or lungs it can go a long distance before dying and be lost.
I vote for the 30-06 in all cases as I do not see the minor improvement in trajectory at distance as anything a normal shoot can achieve (some can, I would need a solid rest and if I had that 5 inch difference on long range would not affect the shot as hold over is hold over and a bit more or less is a matter of practice).
338 is far more than needed though if you are hunting moose or big bear (black bears do not count) then you can use it for dual purpose and poke holes though lighter game than large bears or moose though you better have good shot placement or said game will go dies someplace you cannot find it.