@AlexeyDo: (1) If we already proved that arbitrary small homotopy colimits are preserved by a derived left Quillen functor, then all what remains to be shown is that a derived left Quillen functor preserves finite homotopy limits. This will imply the statement in the main post: a derived Quillen functor between stable model categories preserves finite homotopy limits if and only if it preserves finite homotopy colimits.

@AlexeyDo: (1) Not sure if I understood the question. Small homotopy colimits were treated in the previous paragraph; for any left Quillen functor its left derived functor preserves all small homotopy colimits. (2) You cannot define homotopy colimits in triangulated categories in general. In some very special cases, such as (for example) homotopy coproducts and homotopy cofibers, the notions supplied by triangulated categories (via ordinary coproducts and distinguished triangles) do indeed coincide with the model-categorical notions.

@SimonHenry: If I understood your maps correctly, the vertical morphisms from the first row to the second row would have to include the morphism f, which is not a weak equivalence. For example, in the top row, the first morphism is id:0→0 and in the second row the first morphism is f:0→1, so the first vertical morphism necessarily has to be f:0→1, which is not a weak equivalence and therefore does not yield a legitimate hammock.

@SimonHenry: What is the top row? Especially the two leftmost maps.

@SimonHenry: I fixed the second hammock by adding another zigzag.

Remark 3.2.5 in Riehl–Verity “Homotopy coherent adjunctions and the formal theory of monads” says that the hammock localization of a category with two objects and a single nonidentity arrow is the simplicial category Adj given by the free adjunction.

What is Cat_∞? It is not defined in your post. Also, you seem to claim that (1) and (2) define adjoint functors, but (2) starts with an ∞-category (which presumably means quasicategory here), and it seems that your Cat_∞ means something else than quasicategories. Finally, the construction of (2) as it is currently stated makes noncanonical choices of basepoints, so it is not even a functor, let alone an adjoint functor.

@TimCampion: There is also an explicit description of maps $\def\Ex{\mathop{\sf Ex}}A→\Ex^∞ X$ for a simplicial set $A$ with finitely many simplices: these are precisely the maps $\def\Sd{\mathop{\sf Sd}}\Sd^k A→X$ for some $k≥0$, with an obvious equivalence relation. There is no such simplicial description for the singular complex functor. This is useful in practice, e.g., for the main theorem of arxiv.org/abs/2110.04679.

Moreover, Djakov–Mitiagin (1976) construct a nuclear Fréchet space X such that for any nuclear Fréchet space Y the space X⨯Y has no basis.