It follows from a recent answer that even when a ring is formally étale rather than étale, we can check this condition on localizations, and hence stalks. It's not hard to show that we can define a "formally étale" topology on $Aff$. Presumably there's a good reason for requiring finite presentation), but I can't think of a reason why off of the top of my head.

Specifically, we let the covering families be jointly faithfully flat with each morphism formally étale. This is by construction subcanonical.

Questions: Why are finiteness conditions necessary for schemes in general and étale maps in particular.

What kinds of problems will we run into if we do not put finitness conditions on the "formally étale" topology?

If this topology fails in some way, can requiring that covers are finite families of morphisms (quasicompact), or that the morphisms in the cover are themselves flat, or even both? This would give us a topology similar to the fpqc topology, except in that all covering families would be made up of formally étale morphisms. The only difference between this topology and the étale topology is the finite presentation of the morphisms in the covering families. Does this still not work?