That is an interesting story about Jesse F. Osborne. I would be interested in the source of that information. My father's maternal grandmother was Emma Glenn Osborne, d/o Isaac, a brother of Jesse Franklin Osborne.
I cannot find anything substanial beyond their parents, David Osborne and Nancy Byrom (Bynum). I have seen mention of David's parents being an Edward and Naomi Osborne (no maiden name)in various family trees, but they fail to cite sources.
From those trees, this is what I have gathered:
Edward was born about 1750 in England
He died in 1816 in Franklin County Tennessee
He did leave a will that is recorded in the Franklin Co. Will Book, 1808-1876, Volume 1-2 page 16
He named John Coffee and his wife executors of the estate.
It is pure specualtion that Edward was the father of Jesse F or most of the Osborne's in Franklin Co. TN in my opinion. Until something other than the will is found, I have noted the information not confirmed in my records.
I can share some information I have found in my research about the sibilings of Jesse if you are interested.