Continuity mistake: At the end when Ryan brings the wounded sergeant to the other side of the bridge, Reiben starts to make his turn to provide covering fire for the Captain. He repeats the exact same movement two shots later.
Continuity mistake: When Mellish is stabbed, at first there's no blood on his shirt, but then there is, even before the knife touches him.
Continuity mistake: When Mike crosses the bridge at the end he gets shot and is helped by a soldier. From the German view the bridge is clear, then it cuts to the American side and Mike and the soldier are exiting the bridge to their left towards Miller. A motorbike and sidecar have now appeared behind them and a crouching American soldier is next to the motorbike halfway across the bridge.
Continuity mistake: While they are preparing to defend the bridge towards the end, Mellish tells Upham to be "Johnny on the spot with the ammo." and proceeds to put 2 30 cal. belts around Upham's neck. In the first shot the bullets are pointed down, when the shot changes the bullets are pointed up.
Continuity mistake: When the unit finds the first Private Ryan, after he hears his brothers are dead, he leans towards his superior officer as he cries, but in the next scene he has his face buried in the officer's chest.
Continuity mistake: When James Ryan from Minnesota is told his brothers died, the arm behind him disappears and reappears depending on the angle.
Chosen answer: There's no tree cover to the left. Whoever goes that way will likely be spotted and targeted before the others and get gunned down, but it's their best chance that one of them will make it into grenade range of the nest before they're all killed. It's not a job anyone sane would volunteer for, and the Captain is trying to get someone to volunteer so he doesn't have to potentially order TWO men to their deaths on a mission that all of them, including him, think isn't worthwhile.
Captain Defenestrator