Now the only difference between a 308 and 30-06 is powder capacity.
That's like saying that a 300win mag is inherently more accurate than a 300 RUM.
Both statements are inherently flawed!
More than anything, accuracy is dependant upon the quality of the barrel, squareness of the action, and the abilities of the shooter.
If you took 2 rifles of equal quality, with the same quality barrel, one in 308 and the other in 30-06, both with the same neck and throat dimensions and using the same components for ammo, the only difference between them would be velocity.
Having said all that. It is a known fact that the larger the powder charge a cartridge uses the less efficient the charge becomes. The more efficient the cartridge is the easier it is to tune a load during load developement. The less powder a cartridge uses the easier it is to get each shot (velocity) the same as the last, but the difference between the 308 and 30-06 is virtualy nil. Now when you start talking about the difference between the 308 and a 30-378Wby, you might could say something there, but at the same time, with custom actions and barrels on both the 30-378Wby probably won't be that much more difficult to find an accurate load in.
My personal opinion is that a rifles accuracy potential is made of the sum of the rifles and ammunitions components and how well they were put together, and not in the cartridges powder capacity.