I'm not related but Larkin could have been Joseph's father and the military info about Larkin incorrect.
Larkin C. Taylor
Born: 1802, Burke Co. NC and died 1845, Bradley Co. TN
Married: Martha Read, b. 1808 McMinn Co. TN and d. 1898 Bradley Co. TN
Source: Edge Family 2018b, Ancestry
There is an attachment to this family tree which details Joseph's Civil War service and some of his life.
I'd look for Larkin and Martha's probate records to see if a son Joseph is named there. I'd contact the Bradley County Archives, https://bradleycountytn.gov/archives.html
The Cleveland Public Library, http://tngenweb.org/bradley/library.html
might also have these records. Successful Searching!