@PeterHaine Thanks Peter -- that's a very clever fix! Is there any technical reason that I should take $\mathcal{W}$ to be wide, or is this just convention?

@R.vanDobbendeBruyn Thanks for your comment. I am aware of the set-theoretic issues and usual resolution here. There is also the option to view $\mathsf{Ch}(\mathsf{A})$ as a model category whose weak equivalences are $\mathsf{qis},$ and model the localization using one of the usual methods for model categories. I had in mind fixing some universes $\mathcal{U}\in\mathcal{V}$ (and perhaps beyond this if necessary), and taking all categories to be $\mathcal{U}$-small, so that the localizations would be $\mathcal{V}$-small. Would this not also resolve the issue, or is it more nuanced than that?

The "varieties" mentioned here are the $\infty$-cosmoi: well-behaved collections of $\infty$-categories. Intuitively, we have a collection of $\infty$-categories, and for every pair of $\infty$-categories, an $\infty$-category of functors between them. So, these $\infty$-cosmoi should form "$(\infty,2)$-categories." This is analogous to the fact that the category of categories can be thought of as a $2$-category, rather than just an ordinary category. The homotopy $2$-category is a way of analyzing this $(\infty,2)$-categorical situation by means of ordinary $2$-categories.

Maybe I'm missing something, but I don't see how one passes from the pro-system which is the original definition of the cotangent complex to an element of the derived category of quasi-coherent sheaves on the hypercover. Is this also covered in Olsson's original paper somewhere, or is this something generally known?

This is basically exactly what I wanted. Parts (iv) and (v) definitely do the trick for me. Thanks!