That's right. To put it differently: stabilization is for "big" $\infty$-categories, and $SW_{\Sigma}$ is for "small" $\infty$-categories; passage to $\infty$-categories of inductive limits intertwines the two constructions. The construction $SW_{\Sigma}$ is perhaps more concrete than stabilization because you can perform it at the level of homotopy categories: this is because passage from an $\infty$-category $\mathcal{C}$ to its homotopy category commutes with sequential direct limits, but not with sequential inverse limits.

It seems worth pointing out that the discussion in my thesis contains a mistake. Theorem 3.1.6 is false as stated: if $A$ is a simplicial commutative ring, you generally cannot recover the $\infty$-category of $A$-modules by stabilizing the $\infty$-category of $A$-algebras. Consequently, the theory of derived algebraic geometry (based on simplicial commutative rings) should probably be regarded as a setting where the "categorical" notion of module doesn't work (or at least doesn't recover the usual notion of module), unless you work in characteristic zero or use $E_{\infty}$-rings instead.

I suppose my original answer is misleading: the "Betti" moduli space is defined over the rational numbers (even over the integers), so you could base change it to any field you like. And when you base change it to the $\ell$-adics, you get something closely related to your $\Loc_{n}^{\ell}(X)$. However, the Betti moduli space depends only on the topology of $X$, not on its complex structure. Geometric Langlands is about the deRham moduli spaces, where there is a connection between the algebraic structure on $X$ and on the moduli space.

You could make that definition. However, the business about $\infty$-topoi and higher stacks is a red herring, because $BGL_n$ is an ordinary stack. You also don't get very much structure this way: for example, any $\ell$-adic sheaf $F$ with Zariski-dense monodromy will have an open neighborhood in your $Loc^{\ell}_{n}(X)$ which is isomorphic to BG_m. In particular, $Loc^{\ell}_{n}(X)$ doesn't allow you to "move continuously" between nonisomorphic sheaves with dense monodromy.