I don't think there is any proof that this Jesse Albritton was the son of Enoch and Penelope Frizzle Albritton. This has been published in the Lomas and McSwain books published in the 1970s and 1980s, respectively. It is also regarded as "tradition" that Jesse was the son of Enoch and Penelope.
There is evidence in the records of Wilcox Co AL that Enoch and Penelope had sons Allen, John F., Silas W. A., and George W. Albritton, as well as daughters Levincy and Mary. However, there is absolutely NO evidence in the Wilcox County records, to my knowledge, of a son by the name of "Jesse".
The census records give little help here. Neither the 1820 nor the 1830 censuses indicate any addition sons to the list above. The 1820 and 1830 censuses DO indicate that Enoch and Penelope Albritton had an additional daughter whose name I don't know.
The 1800 census does indicate that Enoch and Penelope had an additional male child born 1790-1800 in their household who could have been an extra son. If correct, this son would have been over 20 by the time the 1820 census was taken, so this suggests they may have had more sons to those proven ones above. But whether or not this possible son was named "Jesse" remains to be seen.
Next, there is absolutely NO PROOF of Enoch Albritton's ancestors. It would APPEAR that he was the son of James Albritton, Jr. and his wife Mary, but so far no one has been able to locate any documentation of this whatsoever. Anyone who has studied the Albritton family in eastern North Carolina to some extent will realize that Enoch is almost certainly a grandson of James Albritton, Sr.; he was closely involved with James Albritton, Sr.'s sons and proven grandsons in Pitt Co NC. But it is not known for certain that his father was James Albritton, Jr.
Further, there is no proof of Mary's maiden name. Many have speculated that it was "Arnett", as this name persists in the descendants of Enoch Albritton. But again, this is speculation and not proven fact.
Finally, the list of Albritton ancestry contains a gap. This list, I believe, comes from Fran Lomas' book published in 1979. In a later work, the late Eleanor Davis McSwain revealed a mistake in this lineage. If anyone is interested in the specifics, let me know.