The Remington M1903 was probably issued to the Army and most saw very little use. The replacement High Standard barrel, however, suggests that your rifle found usage somewhere, or was not well cared for.
And you are correct that early and mid-production M'03-A3's were issued to rear echelon troops and saw little or no combat action. And most late M'03-A3's were never issued.
Good Luck!
J.B.