Personally, I don't think that any so-called "thin skinned" game animal is going to notice the difference between being shot with a .308 WCF or a .30-'06. I've taken game with both. I could hunt happily with either.
Basically, for general big game hunting, if I find a rifle I like in short action rounds from .260 Remington to .308, or standard length action rounds from 6.5 mm Swedish to .30-'06, I don't fret over the caliber too much. I'm confident I can make all of them work for the hunting I want to do and equally confident that no animal I shoot with any round from this broad group is really going to notice what I killed it with.
On the other hand, if the biggest game I ever cared to hunt were blacktail, whitetail, and / or mule deer, I could get by fine with just one of either a .243, 6mm Remington, .250 Savage, 257 Roberts, .25-06, .260 Remington, or 6.5mm Swedish Mauser.
The .30-'06 is my favorite centerfire rifle round, though. But while I can think of all kinds of logical reasons to defend its honorable magnificent greatness, the truth of the matter is that emotion probably plays a bigger role in my favoritism than logic does. When I was most active in "adventure" hunting in multiple states, hunting caribou in Canada, and plains game in Botswana, I did a lot of that with an old peep-sighted Griffin and Howe Springfield that I essentially inherited, and that rifle was -surprise- chambered to .30-'06.
JP