I am an auto type of guy too sj1, accuracy suffers a bit but few of us shoot well enough to have a kill ratio difference between a 1 moa and 1/2 moa weapon...LOL
I like the '06 . Bigger case , more powder equals more down range velocity . Same cal bullets so no SD advantage . For the larger game & longer range the '06 is my vote . For elk & moose in the timber I like my 358Win . Of course it kicks but it'll launch a 225 Barnes or a 250 Original with enough authority & penetration that it even works on Brown Bear .
Because of the following : loaded to a higher pressure to meet '06 velocity , 308 is inherrently more accurate ,shorter case feeds better thru automatic weapons , the military already had weapon systems in that caliber . With a 150 or 165gr bullet it's pretty much a wash . However with heavier bullets it's a different story . I do believe the OP mentioned elk & moose , which in my book means 180gr & heavier . In 60yrs it's been my experience animals are tougher to kill than humans . You've your opinion & I have mine . The ability to state both points in public is the beauty of America .
30-06 was my first rifle and it was great in the 70's and 80's but the surplus ammo is now more plentiful for the 308. The fact that it will cycle thru an AR platform is what has me saving my money for one this year.