I believe the 1903 Mausers are Turkish. Originally they were chambered in 7.65×53, same as the Argentines, so the same problem is likely to apply. Someone ran a .30-06 reamer in, but the pilot, which was probably about .306" to fit military minimum spec .3065" bores, bounced around in the .312" bore, which accounts for the scored chamber, regardless of whether the reamer was sharp or not. The accuracy would be pretty awful with bullets rattling down a bore that loose. It's enough clearance that the bullets could wind up engraved by rifling on one side and not the other. That usually leads to keyholing.
You should slug the bore to double-check the diameter. If you are going to go to .30-06 AI to clean the chamber up, see if you can rent a removable pilot reamer and a 0.311" pilot? Then just neck the .30-06 AI up to 0.311" for seating 0.312" bullets and buy those. Hornady makes a couple for .303 British. If the muzzle has been funneled by soldiers cleaning the bore from that end with steel cleaning rods (common in old service Mausers) then I would replace the barrel. Brownells and a number of the custom barrel makers have drop-ins available for Mausers that aren't too expensive.