In Welty's story "A Visit of Charity," her overall message is that people in nursing homes lose their identity and are treated badly, almost like animals. The many references to animals (particularly sheep) are in the story to show that the elderly patients are treated like animals as they are herded around and their needs are ignored. Welty basically reduces the patients to animals to show what happens when they are mistreated. The patients become interchangeable to the staff, who have no respect for them.
Unfortunately, according to what we read in the news sometimes, there are still cases of elder abuse in nursing homes. While some are good places where patients receive excellent care, some are underfunded or staffed with insensitive people who treat patients with the same herd mentality, denying the individuality and identity they are entitled to. Advocacy groups are working to change that, and it is much better than the time when Welty wrote this story.