I purchased a savage 16 in .308 last summer and really don't care for shooting it because of the recoil. It weighs about 7.5 pounds with a 22 inch barrel. My primary use is target/bench shooting from 200-500 yards. Though I also bought the .308 for the possibility of big game hunting
I see too things here. You have a 308 you want to use for hunting, but the recoil bothers you. But you want a light rifle for hunt ( I certainly do).
The 308 is more then capable for the animals you listed. And plenty well suited for target shooting to 500 yards (I use 308s for 1000 yards and further)
I recommend a muzzle brake. You'll get your 308 down to the recoil of the 223s. You'll have to wear ear protection, which you'd be silly for not wearing with or without a brake.
I built a 308 for my grandson, but the recoil got to him, I put a muzzle brake on it and it works great, he enjoys shooting it and is doing quite will.
I also put one on my 300 WM and my 375 H&H. Call me a wimp if you will, I'm too old to care, I'm more interested in my shooting abilities. I do shoot heavy rifles better with a brake then without.