You could use either, but there are major differences being ignored. That happens when the numbers are the focus, and not the application.
Plinking and some shooting can be done with a single shot .22 cheaper than anything, which points out two issues: Costs, and how much power you really need to project downrange.
Since hunting for live game wasn't the primary choice, then the real focus is shooting paper. Either cartridge will do that job. What is next in line is the power levels - 6.5 is an intermediate, .308 a true full power. There is a significant difference, besides the powder capacity of the cases, there's also recoil, and operating pressures on the guns. It's more than just significant that the 6.5 is developed around the AR15, and the .308 was the intended cartridge for the AR10.
What's being chosen are two completely different guns, not just cartridges. The larger will require getting accustomed to twice the recoil, will weigh at least a pound and a half more equally kitted out, and will cost at least $500 more. It won't be standardized on a milspec platform, as the AR10 makers won't use the same uppers or bolts. Parts are proprietary within groups of makers - unlike the AR15, which you can literally assemble from parts that built on opposite sides of the country.
There's a fact floating around out in the military gun users crowd that some are in complete denial about - intermediate cartridges are the issue power levels nearly worldwide, and full power .30's a special use cartridge. The reason is recoil - soldiers won't shoot big guns as much, and pick up more bad habits dealing with recoil. They also can't control the weapon as well in rapid fire, second shots take longer because the muzzle flips, and the shooter is hit with more force.
.308 isn't the best round for starting out in shooting, any more than going to a pistol range with a .45 as a first gun.
Given what the OP wants to do, the 6.5G is the better and cheaper choice.
