I have a suspicion who this Alexander's father was but I can no more document it well enough to state it as fact than I can prove the father of the Michael Abernathy who married Isabella West on 18 Nov 1826 in Lincoln County NC, despite decades of research on all these Abernathys.
At least with Michael we have one living direct line male descendant whose Y-DNA matches that of the males who can be documented to prove descent from the first Robert. With Alexander, we don't even have that yet, and it does require three for sufficient proof, you know?
So Alexander is just one of many whose paper trail has too many undocumented links for us to do other than speculate. Given the way some folks jump on speculation and mutate it into fact without a shred of an attempt at verification, speculation can be ill-advised.
As long as Abernathys continue to refuse Y-DNA tests I seriously doubt we shall ever get enough data on widely separated cousins to demonstrate via science what is not demonstrated via documents. Baptists in particular seem to object to the tests more than those of other religious denominations.
As for your reference to the Nathan Abernathy also found on the 1820 Burke census, I suspect that he could have been an uncle or father of Alexander, but that does not show which of thesons delineated in the will of Robert 4th Abernathy was, in fact, the father of this Alexander.Even a Y-DNA can't name the names, only show by Y-DNA evidence if a living male is or is not of the family grouping due to the values of his Y-DNA's markers.
Alexander is a given name found in many apparently unrelated branches of Abernethys and Abernathys, but not nearly as popular as William, John, James or even Robert, Thomas, Charles, George, David, or Henry. Nathan is a considerably less popular given name among the Abern?thys. Even Turner, Smith, and Miles are twice as frequently found as Nathan, in my experience.