The 308 is an excellent target or hunting round. Recoil isn't that bad, I've shoot hundreds of thousands of round of 308 with rifles with steel butt plates and I hate recoil.
I don't load hot. I keep my 308s to about 2600 fps MV in 165(8) to 175(180) gr bullets.
Using that load the round still has 1500 ft lbs of energy at 600 yards which is good for elk, more then adequate for deer antelope size animals.
If using a 100 yard zero you need to come up about 13 MOA to get to 600 yards.
There are tons of excellent bullets out there. You want to extend your hunting range look at Berger bullets. They are designed to enter the target a few inches and the come apart, totally destroying the central nervous system and I've shot a lot of game with them, they work as advertised.
Berger says the on target remaining velocity should be about 1800 fps for them to work properly. Using the 185 Gr Berger for hunting, with a MV of 2600 fps, your remaining velocity of 1800 fps would be about 900 yards. Farther then normal hunting ranges. This Berger bullet only needs 10 moa to get you to 600 from a 100 yard zero.
Like I said, I don't load hot. I keep my loads MV at 2600 because I also use the loads in gas guns.
But in your Remington, your MV can be faster, and you'll find factory loads are.
Fear not, you'll be happy with your 308. Any commercial factory load will work hunting. You can get better performance and accuracy by reloading if you want to move to competition shooting.