My main hunting rifle is a Rem 700 Stainless Synthetic Mountain Rifle in .30-06. I would not trade it for the world.
The first three shots of my hunting load with a 168 TSX at a chronographed 3,000+ fps will cut a solid 1/2 m.o.a. off the bench, time and time again. After three shots, the thin barrel heats up and the subsequent shots walk a bit. Big deal. I don't use this rifle for match shooting (I use my Palma and F-Class rifles for that purpose.) For a big game hunting rifle, if you can get consistent accuracy like this for the first three shots, that's all you'll need, and you should be able to shoot animals WAY out there if you have to.
By the way, I have a VersaPod and sometimes mount it on my Remington. It's quickly detachable and doesn't add too much weight or radically alter the balance of the rifle with a sling. Last year I mounted it when we hiked into the back country for a couple of hours and got a Muley buck from 300 yards. I wouldn't hesitate to use it similarly again. The country we were hunting is quite open and rested longer shots are to be expected.
Mr00jimbo, I'd make sure the bedding was good on your rifle and work up a load, probably starting with 165 or 168-grain bullets. I found my Mountain Rifle to be startlingly accurate... for the first three shots. If you are shooting at the range, just let the rifle cool for 20 minutes and go again. It's no big deal if you have other rifles with you at the range.