Many guns would not stabilize a bullet as long as a 168gr VLD in 7-08. Hodgdon doesn't even have load data for a 168gr 7-08. You'd need a twist rate of 1:10 or more, available but not the "standard" in over-the-counter guns.
Let's assume they would though...
The following is from JBM ballistics using a 7mm VLD 168gr hunting bullet at 2,700fps and a .308 VLD hunting bullet at 2,800fps, both speeds are near (or over) real world max velocity for those rounds. Sighted max 3" high, just for reference:
7mm:
-19.1@400
-40.2@500
-69.8@600
-108.9@700
-158.8@800
.308:
-18.7@400
-40.3@500
-71.2@600
-113.2@700
-168.2@800
Since you'd obviously be using range finders and scope adjustments at those ranges, I don't think a few extra clicks really makes any difference. HOWEVER, the 7mm would theoretically be carry 1055ft/lbs of energy at 800 yards, while the .308 is down to 828ft/lbs. That extra 200 ft/lbs of energy COULD be significant. Even so, 800 yards is a shot well beyond the capabilities of the VAST majority of hunters, especially under "real world" conditions.
At 600 yards, there's only a 2.3" trajectory difference and, while there's still almost 200ft/lbs difference, the .308 is still carrying almost 1,200ft/lbs, which is plenty.
600 is still beyond the capabilities of more shooters than not, especially under "real" conditions.
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I'm a big fan of the 7mm-08. Big fan. I think it's darn near the ideal big-game cartridge for North America. I think it's a near perfect balance of recoil, trajectory and energy. It seems to really shine in the "recoil vs performance" category. Most hunters are going to be shooting factory 140gr ammo and many are not capable of shots beyond even 400 yards or so. At those ranges, with factory ammo, the only difference of any significance is recoil, and that's highly subjective.