A big factor in the accuracy of the '03 vs. the Mauser was the quality of ammunition. There were different acuracy specifications for "clipped" or "rifle" pack vs. "linked" or "belted" pack for machinegun use. This was adhered to even in wartime. Rifle ammunition had to produce no worse than a 5-inch Mean Radius at 500 yards, whereas machinegun ammunition could go up to 7.5 inches, unless it was designated as "overhead fire" ammunition for shooting over the heads of friendly troops, which has to meet the rifle specs.
It was also found during WW2 that AP ammunition produced more accurate lots than Ball ammunition, so this was often selected for sniper use. I was told by William C. Davis, Jr. that this wasn't necessarily due to any magic in the design of the AP bullet, but was simply by virtue of the draw and forming dies being new and producing better parts, whereas the Ball M1 and Ball M2 dies were old and by 1943 were mostly worn out.
I still have a quantity of FA34 Ball M1 as well as LC43 APM2, and TW 54 Ball M2 of selected lots which all met the accuracy requirements for Match ammo, being retested at Quantico during the early Vietnam era and producing 3.5 inches or less Mean Radius at 600 yards, being designated as sniper quality and used by the Marines until adoption of the 7.62mm M40. After Vietnam the great majority of that ammo was allocated to the FBI for field office and academy training use until one LTC Oliver North scooped it all up, as well as most of the remaining M72 Match ammo on hand, having it all, well over 3 million rounds, linked up for Browning machineguns, being sent south where it was used with good effect by the Contras.
Larry Gibson, authenticated the story when I told it, saying that he knew people who had the good sense to de-link some of that M72 and repurpose it for sniper use.
73 de KE4SKY In Home Mix We Trust From the Home of Ed's Red in "Almost Heaven" West Virginia