Remembrance We have gone for an in between solution by not declaring things to be either one way or the other, which allows for both paths to coexist. In practice it already is how you propose it to be.
in other words, the model you propose, which we can find, with some differences, in ancient Sparta or in the Zulu under Shaka Zulu, is what inspired the whole Children of Zeta business, which does follow roughly your idea... the Vessels procreate with the purpose of breeding soldiers and workers both for the central authorities (the Palace, The Temple and the Army), and for those who cannot procreate themselves (elder couples, queer couples, single men and women and so on)...Children are recruited/assigned according to their inclination and/or the needs of the city.
Next to this, couples who can procreate and do so are free to either hand over their children to the Temple or to keep them and raise them themselves.
As a player, you are free to decide which way you would prefer things to have gone in the case of Zaton and Zahra.
If in your head-canon all children are passed through the Temple system and are Children of Zeta, that's fine and will probably not be contradicted explicitly in the lore. Potentially the argument can be made that children handed over to the Temple, can be, on request, reassigned to their natural birth parents. We will also not explicitly state that all children go through this process, allowing the player to make up their minds about it.
In other words, it's up to your preference. If incest skeeves you out, then go for the "they are Children of Zeta" route, if not, keep it as is....either way, Ain and Shani regard Zaton and Zahra as their parents, which kind of renders the whole "let's not make it actual blood incest" point a little moot.
As to why incest has become acceptable in this society... it has mostly to do with scarcity of alternatives, with how the few people who started to rebuild and who eventually built Zeta were forced into that or going extinct... we are handwaving the whole incest=genetic defects issue by (at least in my mind) having moved past that particular issue through advancements in medicine and genetic therapy well before the whole End of Days scenario came to be.
In other words, humanity had cured cancer, aids and whatnot else before the End of Days and, incidentally, brought humanity to an evolution point where also inbreeding would not be source of medical issues.... that, at least, is how I personally rationalise the handwave of this particular plothole. (it, combined with several centuries of survival of the fittest, also explains how come most people you see on screen have athletic builds).