SPR is a good flick, but a lot of it is pure Hollywood. The keen eye will recall that the "sniper" character was using 2 different scopes on his rifle at different times in the movie. One of them is a Lyman Alaskan which appears in the photo in post #1. The other is unknown.
I own and shoot a real M1903A4 which I've had for some years. The bore is very lightly used and remains in excellent condition. Remington built them with both 2 and 4 groove barrels which were air gauged for tolerances. Mine is a 2 groove. The rifle will shoot consistent 1.5 MOA groups @ 100 yds using quality handloads with 155 or 168gr match bullets.
The standard scope for the M1903A4 was a standard Weaver 330 sporting scope of 2.5x and with a .75 inch diameter tube. The M84 scope was used on some rifles which remained in service into the 1950s. The Weaver 330 is a very flimsy scope and is highly subject to parallax. It would have been great on a rabbit gun, but really wasn't robust enough for a fighting rifle, even a carefully babied sniper rifle. I have a Weaver M73B1, which is a militarized Weaver 330, but use a Lyman Alaskan on my rifle. The Alaskan is a much better scope and was originally specified for the M1903A4, but Lyman was unable to supply them in the quantities required, so the Weaver became the second best choice. The Alaskan has a 7/8 inch tube and is far superior to the Weaver, but it is still only 2.5x. The post and crosshair reticle subtends approx. 4 inches @ 100 yds, so you can imagine how much real estate it would obscure @ 1 mile.
Most military .30-06 ammo in WW2 was the M2 ball round with a 150-152gr flat base bullet which does not hold up well over a long distance. The 168gr AP bullet was ballistically superior and a lot of this ammo was also used. Long range target shooters know that a 168gr bullet pretty much runs out of gas before 1000 yds. Add this to a very poor sighting system shooting over an unknown distance and it boils down to a lot of Kentucky windage and elevation estimation and a lot of Hail Marys. Is a .30-06 still lethal at a mile;could be, but first you need to hit the target.
Short answer. Yes, it is Hollywood BS.