The box magazine should handle 3.340" cartridge overall length, as that's the SAAMI standard maximum for .30-06, and most magazines have extra room for that plus some. So I suspect something has gone awry with your measuring. My first suspicion is the cause of the confusion is Hornady's poorly chosen name, Overall Length Gauge. That device is used to set overall length
indirectly, but it does not measure overall length directly. Overall length the distance from the outside face of the head of the cartridge case to the tip of the bullet. The gauge measures from the outside face of the head of the cartridge case to the point on the bullet ogive that its insert happens to intercept. That length is used to set up your seating die to turn out cartridges that match what you found with the gauge, caliper adapter and insert. But if you want to know the overall length, you would have to measure your finished cartridges separately with your caliper using no adapters of any kind on it.
A long throat (something Remington is known for) is not an accuracy killer in and of itself. Groups smaller than yours are possible to achieve in a bolt gun with a long throat, so it's not likely the main limitation. Most guns with two-piece stocks and barrel hangers suffer from inconsistent barrel contact points and pressure. For example, since your front sling swivel is attached to a ring on the barrel, if you use a sling and don't get the exact same pressure on it every time, that can cause the issue. You might want to look at M.L. McPhersons work with lever guns, which have a two-piece stock and barrel hangers, too.
Look for any looseness in the contact points with the barrel. Make sure the wood isn't rubbing or contacting it inconsistently. When you shoot for group, remove the sling at the front and try resting the gun's magazine on a sandbag instead of resting the pump stock on it. See if that improves groups.
