@DenisNardin Yes absolutely! In particular, we have a functor F on classical affine schemes (or unanimated algebras) whose left Kan extension (as algebras) takes open embeddings to open embeddings (the classic one does) and commutes with localization. I can say more in email. However, as there aren't resources on derived schemes modeled this way, I want to carefully check the details that this process does define a (hypercomplete) Zariski sheaf.
I agree it can be done, but I'd like to write out precisely how and as this is super well-known to experts, I figured the discussion would benefit others if we had it here (perhaps the wrong chat but you've all been so helpful in the past on these questions). I'm excited for what advise you have Denis, and as always thanks a bunch in advance!
