Actually when your talking about accuracy, all of the above listed rounds perform well. There isn't a slouch in the bunch, when looked at from purely the cartridge side of this equation. Where the devil details are is in the platform, ie: what rifle? All things considered any one of these will give you under 1 MOA in a quality rifle, and with paying attention to details you can half that number or very close.
I disagree about the choice of 7mm bullets available for handloading though. There are a lot of bullets available for the 7mm, and most are not for magnums. There are a few magnum only bullets, most in the "heavy end". Right now Nosler is making partitions, Sierra makes gamekings, Speer does Grand Slams, Hornady makes the interlocks, this doesn't include the smaller outfits like Barnes, or the foriegn ones like Norma. There are piles of good bullets for the slower 7's, I have used almost every manufacturer I listed on game in a 7x57. I really only use two loads anymore for hunting, both absolutly perform on game, and my handloads are giving me right at .65-7 MOA from the bench, with a 20" tube. In my mind it just doesn't get any better than that for hunting and I quit experimenting and fiddling a long time ago. My "Magic" bullets? Nosler 140gr and 150gr Partitions. There are a couple from the above list I can tighten my groups up with, buy cheaper, or get a little more velocity with, but none put it all together as well and give me the performance on game. I also grant that my rifle is niether a stock rifle, and it is not picky on bullet types, it is very cosmopolitan in that regard.
The 7mm platform really is only bettered by the 30 cals in bullet availability. The 6.5 doesn't have the bullet selection of the 7's but what is out there is very good, do some homework, the selection is OK, but nowhere near what you can find in the 7mm's. I load for both 6.5 and 7mm and it is always easier to find my 7 components, I end up ordering most of my 6.5 stuff.
The 6.8 is a 27 cal, and the current one being fiddled with for the AR-15 platform is not even in the same class as any of the above mentioned rounds. It is a very light 27 cal bullet, with nowhere near the powder capacity as these rounds. My comments still stand on the AR-10/FAL use of the 260 Rem, but it must be in a rifle that was chambered for a 308 originally.
I would bank all the comments you have recieved and look for the rifle, this always seems to make the final influence when I am buying rifles. Finding a 6.5x55 in anything but a CZ right now is going to be tough, the 7x57 is difficult to hunt down in new rifles, the 260 Remington has bettter availability, and just about everyone chambers the 7mm-08. So without significant effort on your part your choices become limited, and your first choice of calibers becomes the reality.