Yes, they would. There are documentaries about this, My grandmother was in an orphanage in Greene County Tenn while her mother was very sick along with some siblings. When her parents went back to get them they had adopted her out. Search Re: Tennessee Children's Home Society was an orphanage operated in the state of Tennessee during the first half of the twentieth century, and is most often associated with its Memphis branch operator Georgia Tann as an organization involved with the kidnapping of children and their illegal adoptions. Tann died in 1950 before the state of Tennessee could release its findings on her activities. A story reported by 60 Minutes in 1991 renewed interest in Tann's black market adoptions, and the help she received from Shelby County Juvenile Court Judge Camille Kelley.
So the cute little ones were adopted out to rich barren parents all over the US. The older children were adopted out to people for farm hands and servants, although I assume the said they wanted them as a son/daughter. And these adoptive parents paid for them.
My grandmother was Cecil Dale Hill, she had quite a difficult childhood.