I have never heard anything bad about a 77, but I have had three 700s chambered in .308, I had the triggers lightened up to 3 lbs, and all three of them were MOA or better out of the box.
The gunsmith, who was rolling his eyes at me because I keep buying the rifles, hitting financial trouble and selling them, asked me why I didn't look at a Savage. So I met someone who had one, with the new accutrigger, and he let me shoot it. Honestly, I wasn't that impressed. I didn't feel nearly as clean of a break as the Remingtons.
If I had to pick one bolt rifle to do everything, I would get a Remington 700 VS in .308, with a middle-grade Leupold 3-10x. (I've had one.) Anything more specific is catering to skills that aren't as polished as I pretend they are.