The link in the OP leads to a thread with an answer by Charles Rezk, who wrote
For each "shape" of zig-zag, there is a "hammock category" for it... whose objects are functors $f\colon Z\to C$ ($Z$ is an abstract zig-zag of a particular shape) such that the backwards arrows of $Z$ are sent into $W$. The morphisms are natural transformations $f\to f'$ which are identities at the ends (and which in the original formulation of Dwyer and Kan are such that the vertical arrows of the transformation must also be in $W$, though this condition turns out not to really be necessary, so it is nowadays often dropped)...Go look at the original Dwyer-Kan paper, or at the book by Dwyer-Hirschhorn-Kan-Smith.
Consider a hammock without the assumption that the vertical morphisms are weak equivalences. It might help to look at the picture here or on page 8 of this pdf. In the first link, since the backwards arrows $X \gets K_1$ and $X\gets L_1$ are weak equivalences, then so is the vertical arrow $K_1\to L_1$, by the two-out-of-three property. However, for a hammock with many intermediate layers, the two-out-of-three property does not imply all vertical arrows are weak equivalences.
Now consider the situation from 3.2 of the book Homotopy Limit Functors on Model Categories and Homotopical Categories by Dwyer-Hirschhorn-Kan-Smith. This is about hammocks coming from model categories. In that context, of a model structure, the authors prove that elements of $Ho M(X,Y)$ are in one-to-one correspondence with equivalence classes of zig-zags of the form $X \gets \bullet \to \bullet \to Y$ (let's call any hammock with more non-composable arrows than this "wide"). Note that, for a hammock of this form, working from the outside to the inside, you could deduce that the vertical maps in a map between zig-zags are weak equivalences, knowing that the backward maps are. And the authors point this out in their 3.2. Later, in 7.7, the authors point out that this argument didn't really need anything to do with the cofibrations and fibrations, so should remain true in any homotopical category that admits a 3-arrow calculus. That's proven in their 11.2. I strongly suspect this is what Rezk had in mind, given the quote above.
Now, note that when you drop the assumption about the 3-arrow calculus, then you cannot automatically reduce from an arbitrary "wide" zig-zag to one of the form displayed above (with two backwards arrows and one forwards arrow). In Dwyer-Hirschhorn-Kan-Smith, 34.2, they take extra care to work with arbitrary zig-zags and for these they do need to assume the vertical maps are weak equivalences, in a map between zig-zags. In the original Dwyer-Kan paper, they do require the vertical maps to be weak equivalences when they deal with "wide" zig-zags.
Now, the natural question is "does every relative category admit a 3-arrow calculus?" And the answer is no. You can read more about this in the following papers: 1, 2, 3, 4. These papers illustrate the yoga of the 3-arrow calculus, and give several examples of models of $(\infty,1)$-theory that do admit the 3-arrow calculus, including partial model categories. If someone hands you a relative category, you can replace it by a partial model category of the same homotopy type and use the 3-arrow calculus there, which is perhaps also something Rezk had in mind when he gave his answer.
