The more I look at this the more it seems a made-up muddle of multiple families. The 1790 census in Union, S.C. lists a man named John Cates with a family of three.
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At the same time in North Carolina there is a John Cates with a wife Emilia/Millie and eight children. This John died in 1793 and left a detailed will. Widow Millie and three children are still there in 1800. "United States Census, 1800," database with images, FamilySearch (
https://familysearch.org/ark:/61903/1:1:XHRD-XD7 : accessed 29 June 2021), Milly Cates, Hillsborough, Orange, North Carolina, United States; citing p. 554, NARA microfilm publication M32, (Washington D.C.: National Archives and Records Administration, n.d.), roll 34; FHL microfilm 337,910.They had an son, John, who was an adult in 1793 but he did not move to South Carolina before 1803. He may be the John Cates who married Hannah Smith in North Carolina in 1811 or that may be a younger John. Some of the children of John and Millie moved to Tennessee. This cannot be the same family.
The Joseph Cates from South Carolina appears to be the man who was born in South Carolina, then lived in Newton County, Georgia and married Elizabeth Hammack there in in 1829. They moved to Arkansas. In 1850 they are there with ten children. "United States Census, 1850," database with images, FamilySearch (
https://www.familysearch.org/ark:/61903/1:1:M678-RF2 : 20 December 2020), Joseph Cates, Freeo Township, Ouachita, Arkansas, United States; citing family , NARA microfilm publication (Washington, D.C.: National Archives and Records Administration, n.d.). Although various web sites say Joseph died in 1845 or 1854, they appear to be the same family living in Locust Bayou, Arkansas in 1860. "United States Census, 1860", database with images, FamilySearch (
https://familysearch.org/ark:/61903/1:1:M87T-9H5 : 18 February 2021), Joseph Cates, 1860.