Flush your brain of all the nonsense that you've heard/been told, take a step back and start at the beginning.... Don't be in a rush to make bad decisions.
Hitting at at any range, let alone "long range" with a rifle is about shooting. People go way out of their way to avoid the one thing that causes success- spent primers. As a smart man has said- most hunters are waaay over headstamped, over scoped, over weighted, and under bulleted.
You need solid mechanicals, a scope that works every single time, good bullets and low recoil. Combine those and you get FUN. Fun= shooting. A 13-14lb 308 isn't the "deer/elk/sheep gun".
First- do you reload? Going to say "no" as you're not a rifle guy. That means good factory ammo that is readily available is top of the list. 223, 6.5 Creemoor and 308win.
The best thing you could do is get a fast twist .223 Remington (Tikka SL, Kimber Montana), a SWFA SS 6x or 3-9x40 scope, and a trunk full of 75/77gr ammo. Combine that with a similar light weight, low recoiling rifle (6.5Creed/260,7-08, 308 with 155's) set exactly the same, shoot until the barrel is almost worn out, then go forth and slay animals.
Heavy, high recoiling guns don't get shot. Light weight, easily handling, low recoil guns with scopes that work get worn out.