This appears to not be true as stated. Barton traveled to England in the early 1870's, but the American Red Cross was founded in 1881. She was eventually replaced as head of the Red Cross in 1904 at the age of 83. Barton had been criticized for her financial management and was accused of failing to properly distinguish Red Cross funds and her own salary. But a Senate investigation cleared her of any wrongdoing. And her accomplishments during this entire period are extremely impressive. Barton died in Maryland.
"In 1865 Barton hired a staff and opened the “Office of Correspondence with the Friends of the Missing Men of the United States Army” in this building. At the end of the war, Barton took up the cause of grieving parents, family and friends whose sons, brothers, neighbors were missing. She responded to over 63,000 letters, most of which required some kind of research that eventually lead to published lists of the names of the missing so that anyone with knowledge of their whereabouts or death could contact her. By the time the office closed in 1867, she had identified the fate of over 22,000 men."
I know she was also involved in locating and marking the graves of many of the men who had died at Andersonville. A former prisoner Dorance Atwater, smuggled out a list of names of the Union soldiers who had died, and when he learned of Barton's efforts to correspond and help loved ones find out what happened to their family members, he contacted here, and they worked together on the project.
Site link:
Clara Barton's Missing Soldiers Office
Here's a Washington Post article about it from this past April: