Did the show actually say the embryos were next-generation Augments created by Khan and his contemporaries? I remember the impression I walked away with was that they could've been leftovers from the original Chrysalis project in the '70s, so they were Khan's lost siblings, and not lost children.
I believe it was said that the embryos were captured by government forces when the Wars ended. This, to me, would strongly indicate that they were engineered by Khan or his contemporaries.
That wasn't specifically stated. All that was said is that they were "left over from the Eugenics Wars" and that they weren't destroyed after the Wars because it was too controversial. It could be that, in the wake of the Eugenics Wars, the governments of the world investigated what had led to those wars and discovered a hidden Chrysalis cache as a result, leading to a controversy over what to do with them. Thus, people looking back on history could easily blur that together with the Wars themselves. People rarely remember the precise details of everything they studied in history class.
And as I said in another thread, the death toll given in ENT of 30-35 million casualties would seem to indicate that the Wars were fought all across the world, on the scale of a proper, full-blown world war, not scattered brushfire conflicts fought here and there.
Well, as I've said before, I think it's possible to interpret the EW novels as consistent with an open conflict, because it doesn't go into detail about the coverups.
And here's a thought: why do you suppose they were called the Eugenics WarS, plural? Maybe the 1993-96 conflict was only the first Eugenics War, and others came in the ensuing decades.