Liz, the comments from Hans didn't provide specifics, but a little Google Fu found what I believe is the source of his information: https://www.familytreedna.com/groups/mullins/about/results.
Unfortunately, the last update to that page at the FTDNA Mullins DNA project was October 2015 where it noted that, "All updates (other than the basic results charts) going forward will only be housed in our new website." That website was at WorldFamilies.net, which closed shortly after implementation of the GDPR in 2018. The last Wayback archive of the defunct URL linked by the project was 23 May 2018, and I'm afraid it contains no other definitive information, only one test-taker's speculation about the potential NPE.
That "results" page at the FTDNA Mullins project is lengthy, and needs a close read to help inform further research. One seeming disconnect that jumps out is the commentary's reference to Booker Mullins Sr, alternatively referred to as Old Booker Mullins, but the dates for this individual are shown as 1768-1864 (see also https://www.findagrave.com/memorial/210096326/booker-mullins_(adkins)), while Mullins-773 shows as being born c. 1805 and dying after 1880. To me, this points to the Booker Mullins, supposed son of William Mullins, profiled at https://www.wikitree.com/wiki/Mullins-795. This was a brother of William Mullins-314, not a son. However, the dates are still in disagreement with the FTDNA commentary: Booker Mullins-795 is shown at WikiTree as having been born in 1778 and dying before 1 April 1867. Those dates, though, are much closer to those shown for Old Booker Mullins at FTDNA.
Further, there's this excerpt from the FTDNA project:
"All of the men who match with the descendants of Booker Mullins also match an individual who apparently descends from one Henry Atkisson (a.k.a Atkinson or Adkin/Atkins) born circa 1687, and died circa 1730, who was living in one of the parent counties of Lunenburg, and who apparently had at least two sons who (it is believed) resided in Pittsylvania Co., VA."
Could the purported NPE have been associated with William Mullin's brother, not his son? Or do we have various bits of information conflated between the two?
The text at the FTDNA project also mentions a "1936 letter indicating that Booker Mullins was genetically (on the male, or Y-chromosome) NOT a Mullins." But it, unfortunately, provides no citation that would help us locate that letter. It's described on the "goals" page of the project this way:
"Two of our original participants both descend from Booker Mullins (b. ca. 1767--d. ca. 1864) through two of the sons of Sherwood Mullins (1790-1881), son of Booker. Although by all accounts (of existing records, and family history) Booker Mullins appears to be a member of the family of William and Elizabeth Mullins of Pittsylvania Co. (later, Franklin Co.) Virginia, a letter written in 1936 by a grandson of Sherwood Mullins (son of Booker) cast doubts on this relationship from a Y-DNA standpoint, and indicates that the genetic father of Booker Mullins was an Adkins."
The data table of yDNA results at the FTDNA project seems, overall, to correctly group any mentions of Booker Mullins into "Com. Adkins (I-A22208) Ancestor" (other than one Booker c. 1841-1993 that is shown in the "Matt Mullins" section with a different basal haplogroup). We know that the brief, earliest known ancestor information at FTDNA is input by the test taker and that there is no vetting or research done to substantiate it unless the project admin offers details, but all mentions to Booker Mullins seem to correlate with William's brother, not his son. I say "seem" because I think a deeper dive into the research is merited.
This was only a cursory glance, so trust nothing that I say. 
But pending additional information, I agree with you about not removing Mullins-773 as a child of Mullins-314 at this time. The yDNA may tell the tale, but I think there could be conflated information between Mullins-773 and Mullins-795...possibly both the reported DNA correlations as well as some of the genealogical evidence.
Good luck! 
