Step 1: Reduction to a comparison of simplicial spaces.

It suffices to show that the map of simplicial categories

$wS_\bullet E(\mathcal{C}) \xrightarrow{(s,q)} wS_\bullet \mathcal{C} \times wS_\bullet \mathcal{C}$

induces a homotopy equivalence on geometric realizations. We use the simplicial category $\operatorname{Ar}(wS_\bullet \mathcal{C})$ (the arrow category, i.e., morphisms in $wS_\bullet \mathcal{C}$) as an intermediary.

Step 2: The arrow category. Define $F_n \mathcal{C}$ as the category whose objects are sequences of cofibrations in $S_n \mathcal{C}$:

Objects of $S_n E(\mathcal{C})$ are "filtrations of cofibration sequences": $\{A_{i,j} \rightarrowtail B_{i,j} \twoheadrightarrow C_{i,j}\}_{0 \leq i \leq j \leq n}$ where $(A_{0,\bullet} \rightarrowtail B_{0,\bullet} \twoheadrightarrow C_{0,\bullet})$ is the primary data.

This factors through two intermediate categories:

$wS_n E(\mathcal{C}) \xrightarrow{\alpha} wF_n \mathcal{C} \xrightarrow{\beta} wS_n \mathcal{C} \times wS_n \mathcal{C}$

where $F_n \mathcal{C}$ has objects consisting of chains $A_{0,\bullet} \rightarrowtail B_{0,\bullet}$ (just the cofibration, without the explicit quotients).

Step 3: The functor $\alpha$ is a homotopy equivalence. The functor $\alpha$ forgets the explicit quotient $C_{0,\bullet} = B_{0,\bullet}/A_{0,\bullet}$. But in a Waldhausen category, pushout along a cofibration exists and is unique up to unique isomorphism. So the "quotient data" is determined by the cofibration $A \rightarrowtail B$, and $\alpha$ has a (homotopy) inverse that constructs the quotient. More precisely, the nerve of the fiber of $\alpha$ over any object of $F_n \mathcal{C}$ is contractible.

Step 4: Analyzing $\beta$. The functor $\beta: F_n \mathcal{C} \to S_n \mathcal{C} \times S_n \mathcal{C}$ sends $(A_{0,\bullet} \rightarrowtail B_{0,\bullet}) \mapsto (A_{0,\bullet}, B_{0,\bullet}/A_{0,\bullet})$.

We need to show $|wF_\bullet \mathcal{C}| \simeq |wS_\bullet \mathcal{C}| \times |wS_\bullet \mathcal{C}|$.

Step 5: The "path space" trick. Define $P_n \mathcal{C}$ to be the category with objects being filtrations:

$* = X_0 \rightarrowtail X_1 \rightarrowtail \cdots \rightarrowtail X_n$

(same as $S_n \mathcal{C}$, but we think of it as the "total" filtration). And define $\tilde{F}_n \mathcal{C}$ by:

Objects: pairs of filtrations $A_\bullet, B_\bullet \in S_n \mathcal{C}$ with compatible cofibrations $A_i \rightarrowtail B_i$ for all $i$.

The projection $\tilde{F}_n \mathcal{C} \to S_n \mathcal{C}$ via $A_\bullet$ has fibers that are equivalent to $S_n \mathcal{C}$: given the "base" filtration $A_\bullet$, the "total" $B_\bullet$ is determined by its quotient $C_\bullet = B_\bullet/A_\bullet$, which is an arbitrary object of $S_n \mathcal{C}$.

Step 6: Simplicial identities. By the realization lemma for bisimplicial sets, it suffices to show for each $n$ that:

$|wF_n \mathcal{C}| \xrightarrow{\beta_n} |wS_n \mathcal{C}| \times |wS_n \mathcal{C}|$

is a homotopy equivalence. This follows from the fact that $F_n$ is the "twisted product" of two copies of $S_n$, and the twist (the cofibration relation) is contractible.

Formally: the functor $\beta_n$ is a Quillen fibration (satisfies Theorem B conditions) with contractible fibers. The fiber over $(A_\bullet, C_\bullet)$ is the category of cofibration sequences $A_i \rightarrowtail B_i \twoheadrightarrow C_i$ compatible with the filtrations. By the extension axiom of Waldhausen categories, such extensions exist and form a contractible category (any two extensions are connected by a zig-zag of weak equivalences).

Step 7: Conclusion. Combining Steps 3--6:

$|wS_\bullet E(\mathcal{C})| \xrightarrow{\alpha} |wF_\bullet \mathcal{C}| \xrightarrow{\beta} |wS_\bullet \mathcal{C}| \times |wS_\bullet \mathcal{C}|$

Both $\alpha$ and $\beta$ are homotopy equivalences, so $(s,q) = \beta \circ \alpha$ is a homotopy equivalence:

$K(E(\mathcal{C})) = \Omega|wS_\bullet E(\mathcal{C})| \simeq \Omega(|wS_\bullet \mathcal{C}| \times |wS_\bullet \mathcal{C}|) = K(\mathcal{C}) \times K(\mathcal{C}).$

$\square$