At over 800 yards the 30-06, with its higher BC bullets, has a clear advantage...I wont argue that.
But up to 800 yards the 308 can hang right in there with the 30-06...and, more often than not, beat it. (with the same guy doing the shooting)
It does not shoot quite as flat (fast), but trajectory is the easy part...wind is the real challenge, but that too can be "figured out" and is easily repeatable (with practice)
I have no idea which is better or more accurate...just that I do my best shooting with a 308, no matter the range.
I don't have a 30-06 set up for LR shooting...but I do have a few 308's and 2 300 Win Mags.
The Win Mags are accurate and will easily reach the 1,000 yard mark, and then some, but I am more accurate shooting the 308...even at 1,000 yards.
I'm guessing its more me than the round itself...as I get older, I don't care so much for recoil (even in 13 pound rifles....less is better)...which leads me to shoot the 308 MUCH more than the 300...
And therein lies the answer to this question...get familiar with whatever you choose (know the limits of the rifle and ammo), and the little guns can hang in there with the best of them....a 308 at 800 yards can do just as good as any other round...that does not mean that it will at any given time.
And add a lil moly to the 308 bullets (208 AMAX's) and watch that lil booger "step right on out there" to well over 1,000 yards.