The M16 sling was the same as the M1 sling. Only fairly recently, particularly with the M4, has it changed to a low-ready carry (easily). When I was in Basic Training, back in 1998, we used M16s, but had to carry them at the low-ready anyway. That light rifle got to be a rather heavy brick after 12 or so miles of humping.
The new sling setup for the M4 is MUCH better.
Oh, and the scene where being referenced here is lame. No Infantry squad would walk through enemy territory in a wedge formation and carry on a conversation. Hollywood nonsense, but it had to service the plot.
Other thing in Private Ryan is that there’s one B.A.R. being carried as well.
Everyone walking in “one grenade takes us out” groupings (easier to film, easier to converse) is endemic to all movies and TV and I don’t see that ever changing, though I think Black Hawk Down was much better in that respect.