I see what may have been confusing you. As I said (p. 136 original version, 134 of the second edition), you can start with cochain complexes $Y$ and $Y'$ that don't necessarily arise by dualizing chain complexes, and then the Kunneth theorem does apply directly. The finite type question arises from the dualization.

Agreed, but it is always fun to disagree with you :)

It's a fair question, and I apologize for not answering sooner. Hard not to give an unduly long response. Even model categories are rarely directly at the heart of calculations, but they can streamline them, and they can set up structure that can lead to them. For example, localizations of spaces at homology theories are constructed model theoretically. Knowing they exist sets up a wealth of things calculated in modern homotopy theory. A not too sophisticated source leading up to that is ``More concise algebraic topology'', by Kate Ponto and myself. It is meant to introduce the general idea.

I just saw the comment about Bredon. He introduced ordinary $\mathbf{Z}$-graded cohomology theories, in 1966 I believe. (I heard him talk about it that year). Most people outside core algebraic topology mean Borel cohomology, not Bredon cohomology, when they say "equivariant cohomology'', and Borel certainly came earlier. Of course, neither considered $RO(G)$-grading.