I'm wondering if maybe the kids were hidden in a separate part of the Botany Bay with shielding that the Enterprise was unable to scan through, but I doubt that 20th-century (or 21st-century in the rewritten timeline) shielding methods would have been able to fool 23rd-century sensors. And of course Kirk would've had to find out when they reached Ceti Alpha V.
But it is pretty much necessary to explain all Khan's twentysomething followers in TWOK, 15 years after being stranded (or 18 years per the Chronology's inexplicable retcon that even Michael Okuda didn't remember the reason for when I asked him about it). At least, it's more straightforward than Greg Cox's handwave that they were born on Ceti Alpha V and matured at an accelerated rate (and it avoids the ramification that the crew that got blown up on the Reliant would've been mostly underage).
The jump from 15 to 18 years was likely to keep the events of TWOK closer to The Final Frontier, which has to be set 20 years after Day of the Dove, approximately?
That's my assumption about things, anyway. To shorten the gap between films.
Also, Generations made it very difficult to fit TWOK before 2284.