I don't know enough about Marshall's personal character to comment specifically, but I would argue that the Saving Private Ryan premise — which has some basis in reality — goes beyond simple sentiment; it's actually got as much to do with public relations and public opinion on the war.

Since 1948, the Department of Defense has operated a (slightly misleadingly-named) 'Sole Survivor Policy' which allows for serving siblings of US military personnel killed in action to be discharged on compassionate grounds. The policy has its roots in the public outcry over the deaths of the five Sullivan brothers when the USS Juneau was sunk in 1942. The film was loosely based on the Niland Brothers.

It depends on your view of the strategic importance of public relations at war, but I think the WW2 political and military command in the US was well-apprised of the need to cultivate public opinion around what was an increasingly long, expensive and bloody war. I don't have too much trouble believing this kind of mission could be made a political priority — and generally, once the politicians have spoken, these things have a way of becoming a priority for the military command. Would Marshall have dedicated an entire division to recovering one man? Probably not. But sending half a dozen men to find a sole survivor? I don't think that's beyond the realms of possibilty.

As a followup to my first comment: I just came across this lecture, entitled A Case Study in Principled Leadership: General George C. Marshall’s Core Beliefs, which provides some interesting insight into Marshall's character (while arguing that Marshall's personal character has been chronically under-examined.) This section in particular seems relevant to your question (emphasis mine):

Following Pearl Harbor and U.S. entry, Marshall wrote letters to the parents and wives of all the soldiers who were killed. He had to stop doing that—the combat deaths were too many—and he resorted instead to an engraved card. The signed letters he did promptly write—in cases where two or three members of the same family were killed—make hard reading even today. Marshall made sure that President Roosevelt received detailed weekly reports on casualties.

The human cost of the war was brought home to his own family when he received a radio message on May 30, 1944, from General Mark Clark, commander of the Italian campaign. It informed Marshall that his stepson, Allen Brown, the 27-year-old son of his wife, Katherine, had been killed in action the previous day near Compoleone, as his tank unit advanced north toward Rome. A devoted stepfather, Marshall was devastated. It was a week before D-Day in Normandy. Later Marshall visited Allen’s grave at Anzio, talked with members of his stepson’s crew, studied the terrain, and worked to reconstruct the action surrounding Allen’s death.

If that assessment of Marshall's character is accurate, then perhaps the Saving Private Ryan scenario wouldn't necessarily be out of character at all.