1 turn in 10 inched is the standard twist rate for the 30-06 and will stabilize bullets from 110gr to 220gr reliably. The reason many old military rifles shoot heavier bullets better is because they generally have longer throats than commercial sporting rifles. Plus they've had who knows how many 1000's of rounds fired through them over the years and the throats get worn even longer. Lighter bullets generally have shorter bearing surfaces than the heavies of the same design, if the bullet leaves the case neck before it engages the rifling it can sometimes go a bit squirrely and actually travel down the bore slightly crooked causing the bullet veer off course in flight. Clean the bore thoroughly until all traces of fouling are removed and try some 180 or 200gr flat base bullets. If those don't do any better you could have a stuffed barrel.
Counter boring can also cause accuracy issues if it's too deep and/or off center. I normally won't buy a rifle that's been counter bored unless it's an uncommon variant. I've had several Finn Mosin Nagants over the years that were done this way and they just didn't shoot as good as those that weren't.