Your info sounds pretty convincing to me! I'd suggest then the Greene county statement might be in error, though we are not likely to ever know for sure. Civil War record keeping in the the west was hardly complete accurate and thorough. :>)
I'll dig a liitle bit more on the Eli Clay conscript record and try to find out what if any tie to William S Clay is. I did note that William S had a son Ellis El(i)y so I expect there is some family connection.
I find it interesting to note that the Clay family just disappears from the biographical histories of St. Francois county after the war. Another "soft sign" that they were more strongly aligned with the South than average.
John R