I've been pleased with the nearly 10 year old CZ550 Varmint rifle in 308, walnut stock. I like the detachable box magazines, as one can purchase 10 round magazines which at once are easier to remove and replace than the stock 4 round magazine, and offer greater capacity. These magazines are very high quality heavy gage metal.
Once you have worked out your come ups and hold offs for wind you can turn in a respectable performance to 600 yards. Having seen KraigWY shoot his M14 at 600, I can tell you with no reservation that the 308 is an amazing cartridge in the right hands. If memory serves, KraigWY is a NRA High Master, and when he speaks I listen. Attentively. I suggest OP to take his advice to heart, as he truly does know whereof he speaks. I hope I have not made him uncomfortable tooting his horn, as he is not a braggart by nature, but jovial, approachable, and helpful.
Elsewhere, and maybe only tangentially relevant, the "belly bench resters" in F Class shooting enjoy either F "open" or FTR classes, the latter of which is limited to 223 or 308 cartridges IIRC. The "open" shooters will often shoot some amazingly slick bullets at very high velocities, in effort to minimize the effects of wind on their shots out to 1000 yards. Yet the FTR guys will often outshoot them using the "inferior" 308 at that same distance.
After all other factors have been eliminated, it is more about the nut behind the bolt than anything else, his ability to read wind and hold his rifle consistently, shot after shot.