I definitely don't know everyone in algebraic topology, but it seems like pretty much everyone working in stable homotopy theory only works with simplicial sets or CW complexes.

It may be that the induced topology is just not subcanonical, since it would be quite strong (stronger than fpqc I think).

Yeah basically. You'd need to at least prove that given a pure morphism $Spec(R)\to Spec(S)$ and a map $Spec(R)\to Spec(T)$ such that the pair of pulled back maps $Spec(R\otimes_S R)\to Spec(T)$ agree, you get a unique map $Spec(S)\to Spec(T)$.

Also just found this. Section 8 seems to indicate one needs a "centrality" condition, but perhaps you are working with commutative rings anyway. arxiv.org/pdf/q-alg/9707022.pdf

@MarcHoyois ah okay. Yeah, I think I do know how to do this in that case, since you can iteratively build the Amitsur complex. I recall Clark Barwick saying at some point that there's a unique map from the free monoidal category with an algebra to associative ring spectra that picks out the Amitsur complex, and that it's unique. But I don't know where this is written down.

Thanks @MarcHoyois do you know to what degree this holds for, say, ring spectra, commutative or not?