It looks like I completely missed this.  

Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets):  If $X$ is a simplicial set, and $\mathcal{P}$ a finite collection of subobjects of $X$ which is *closed under intersections*, and the $\bigcup_{K\in\mathcal{P}} K=X$, then $\mathrm{hocolim}_{\mathcal{P}} K \approx X$. I think the original non-proof of 2.19 that I gave was really trying to prove this weaker version (I'll try to get that straight when I have some time).

Now the weak version of 2.19 applies to the poset $\mathcal{P}_K$.  Almost: $\mathcal{P}_K$ isn't necessarily closed under intersection (finite limits), but only because it fails to contain an initial object, corresponding to $\varnothing \to K$.  If you append this object to $\mathcal{P}_K$, the argument seems to go through (note that $V_\varnothing(A_1,\dots,A_m)$ is itself empty).