Tennessee officially started recording births and deaths in 1908 except for it's four major cities. However some counties did record them earlier.
I'd check with the Rutherford County Archives, www.rutherfordcountytn.gov/archives/index
to see when they started recording births and deaths. Those dates look too early to me but every county is different.
The Rutherford County TN Historical Society, http://rutherfordtnhistory.org/
has very reasonably publications.
Also a list of resources for the county at https://rclstn.org/genealogical-resources
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