I'll admit that I don't remember every detail of the movie although I've seen it 3 or 4 times over the years, but I'm not trying to start a discussion about movie trivia.
Additionally, why did this group of people have to trek halfway across France? Couldn't they just like....radio the platoon leader and get Private Ryan sent home on the next flight back to the states?
Private Ryan was "inspired" by the story of Fritz Niland.
Fritz was in the 101st airborne division and parachuted in as part of D-day, just like the Ryan character. He was the youngest of four brothers. His oldest brother, Edward, was shot down in Burma in May 1944. Edward actually survived but was marked MIA assumed dead. Fritz's other brothers both died on June 6th - 7th, during the early Normandy assaults. When this was realized, Fritz was sent stateside where he served the rest of the war. There was no official sole survivor policy 1944, but there was beginning to be an informal one.
That said there was no special mission of "Saving Sargent Niland." The order went out to recall him to the US via normal channels.
However if someone wanted to talk to a specific solder in the 101st on June 7th, something like the movie's mission might actually have been necessary. Normal channels were a complete mess for a few days. The 101st as an airborne division was dropped behind enemy lines. The night landings had been chaotic and troops were dropped in some cases 20 miles or more from where they were supposed to be. It took a week or so of heavy fighting for the two airborne divisions to really reorganize and get back into order. No one could find a putative Private Ryan by calling his platoon leader, because no one would have known where an individual platoon was. You could have radioed division headquarters, but even a couple days in they didn't know where most of their individual platoons were. Most airborne troops were fighting in ad hoc units that were formed with the guys who had landed near them, rather than their actual units. As days passed these units reformed into their actual proper units with the right men under their actual officers. But on the 7th finding a specific enlisted man would have been really hard.
In Fritz Niland's case he was able to rendezvous with part of the 501st parachute regiment, his actually correct unit. On the first day the regiment was only 100 or so men instead of the couple thousand it was supposed to be. So calling him should have been easy. He was actually where he was supposed to be. As virtually no one else was, apparently no one did call him. Oddly enough, he didn't get his orders home until he went to the 82nd airborne division HQ. So few men were where they were supposed to be, that the orders had been sent to both division commanders just in case he showed up.