It's not the cartridge itself that produces accuracy...
A well-made 7.62x54R rifle fed good ammo has the potential for very accurate groups.
Likewise, a well-made .308 rifle fed good ammo also has the potential for very accurate groups.
Start throwing variables in like sloppy chambering, oversized bore, poor bedding, heavy and creepy trigger, milsurp ammo, and so forth - then the accuracy potential declines, regardless of chambering.
Where some calibers really shine is when the rifle is matched perfectly to the cartridge it shoots. Cases in point include the 6.5x55 Swedish Mauser M96 rifle, which would probably be impossible to produce these days due to labor costs. Same goes for the current batch of K-11 Swiss Schmidt-Rubin rifles and their 7.5x55 round. Both rifles and their respective rounds exhibit a very high level of quality.
Compare that to, say, Pakistani-made cordite MkVII .303 British ball ammo, as mated to an Ishapore NoIMkIII SMLE. Sloppy ammo, sloppy rifle, anybody's guess as to how it'll group. Swap the Ishapore with a British-made NoIMkIII SMLE in good condition, and chances are the groups will get tighter. Then swap the Pakistani click-bang ammo with good Greek or S&B fodder, and watch as the groups get tighter yet.
In other words, it's not the cartridge, it's the mating of the cartridge to the rifle, specifically the rifle you plunk your money down on.