@GregorSamsa: the homomorphism constructed by Jeremy is not an epimorphism in general. Hungerford supposes that the $h_i$ are generators of $H$, which is not a hypothesis in what Jeremy wrote above.

@TylerLawson but the 2-sided bar construction which you mentioned above only computes Tor when $R$ and one of $M$ or $N$ are flat over $\mathbb{Z}$, no?

Yes, this is fine. If you unravel what the isomorphism map is, you'll find it is the sum of maps $CH_i(X)\{i\}\to M^c(X)$ which are the adjoints to maps $CH_i(X)\to Hom(\mathbb Z\{i\},M^c(X))$ which take the class of a subvariety to the motivic Borel--Moore fundamental class of it in $X$. It suffices to check this is natural for proper maps, and it is, essentially because proper pushforward for Chow groups and for motives with compact support are defined analogously (using the degree of $f$).

@DavidWhite I was a bit hasty in my previous comment. This question is not about flatness of cofibrant S-modules (for which I give a reference in the original post), but about flatness of cofibrant commutative S-algebras, of which the reference you give seems to make no mention of.

@DavidWhite You're thinking of the sentence "Although the we do not know of a general principle that would imply the second property, it holds in all presently known monoidal model categories of spectra [3,4,9]" (where [3] is EKMM)? It's encouraging that the authors would claim so, but do you know where's a proof?

If you have a countable sequence of maps in a triangulated category, you can build an object which you can meaningfully call the homotopy colimit. Taking this approach in the setting of symmetric spectra, you can read a proof of the result in Proposition II.5.12 in Schwede's Symmetric Spectra book project.

An arXiv preprint from today, "Combinatorial model categories are equivalent to presentable quasicategories" by Dmitri Pavlov, is relevant here: arxiv.org/pdf/2110.04679.pdf