I haven't been able to find out anything more, but I also haven't had time to go down there and put boots on the ground.
My next line of approach is to try to find Elizabeth's grave. I know where Jesse and his last wife are buried, and I suspect Elizabeth and others would be nearby.
I did find the Death Certificate for one of the kids.Unfortunately, the Mothers Place of Birth was not listed.
Bates, Phillip H. No. 19066 Place of Death County McLean Inc. Town Rumsey Full Name Phillip H. Bates Male White Widowed Date of Birth March 1845 Age 69 yrs. 3 mos. Occupation Farmer Birthplace Muhlenberg Name of Father Jesse Bates Birthplace of Father Virginia Maiden Name of Mother Miss Arterberry (Elizabeth) Birthplace of Mother Unknown Informant O. T. Payne Date of Death 2 July 1914 Place of Burial New Cypress, Sacramento, McLean Co, Ky Date of Burial 3 July 1914