It could work out, if previous generations had children late in life.
Or if "great-grandfather" is accepted as shorthand for all values of n in n-times-great-father. Which in practice would hold at least for n>3: people get tired of saying "great-great-great..." or at least risk sounding awfully comical if they do so.
If anything, soldiers might be suspected of having children early in their lives. Would others in the Archer family have been military men?
There aren't any black holes in the known Solar system.
And no known means of establishing that Voyager 6 would have fallen into one if one was
here but was atypically invisible. That is, unless the probe fell into an insystem black hole (or, rather, a classic Star Trek wormhole mistaken for a black hole by 20th century science) well within sight of Earth instruments or fellow probes.
We actually had an insystem black hole in DS9 "Past Tense", plus other insystem spacecraft-sucking anomalies in "One Little Step" and (by implication) in the "backstory" of ST5:TFF.
Then again, they were called the Eugenics Wars, plural
...By Spock. He would probably readily include WWII in there, as from the 23rd century alien viewpoint eugenics would be the one outstanding theme in that particular disaster!
I'm not really a fan of Greg Cox' "belittling"
of Khan's history, but the idea of counting many real-world conflicts from the late 20th century as "Eugenics Wars" is perfectly acceptable as such. We can dream up additional, wholly fictional yet wholly realistic conflicts of that nature, too, to give the plural full justification.