The .308 just doesn't perform consistently for me past 500 yards
Who told you that, I've been shooting 308s in competition for over 35 years. 200, 300, 600 and 1000 yards, and it DOES WORK.
I do have a 300 Win 1000 yard rifle, it works but fatigue will get you after a while. Actually the best 1000 yard scores I've show were with my M1Z and 308s.
Besides target shooting, I also plan to use the gun for hunting elk, bighorn sheep, moose and bear.
The 308 is more then adequate for those animals. People go to magnums thinking its about power, NOPE, its about marksmanship.
If you just want another rifle, then "want" is reason enough. But for what you've listed, the 308 is great.
The 6/6.5 MMs are great also, but for the larger animals you talk about a heavier 308 would work better in my opinion.
The thing is if you shoot any of the animals you mentioned in the heart/lung area, more then likely the bullet will exit the other side. So will the magnums, but how far past the animal do you want the bullet to go after shooting it.
The magnums are ok, but for hunting they require a heavier gun then I like to carry while hunting. As for target shooting, again, they will work, but recoil and fatigue will take its toll and effect your shooting.
Again, I'm not against buying another rifle, heaven knows I have my share, everywhere from 25-20 to 416 Rigby, but the target rifles I use now are either 223 or 308 ('06 in Vintage Military Rifle matches) and my main hunting rifles are a 257 Roberts and 270 Win.
Just my humble opinion.