Why do we care about $(\infty,2)$-categories? crossposted from MSE as suggested by Igor Sikora
Homotopy theory provides much motivation for studying $(\infty,1)$-categories in their relations to homotopical algebra, derived geometry, stable homotopy stuffs, cohomology, physics, and so on. As for $2$-categories, one doesn't even have to motivate them since they're all over the place.
However, I'm having a hard time motivating myself to study $(\infty,2)$-categories. I've been learning a bunch of facts about them: how the Duskin nerve can be regarded as an embedding from bicategories to the complicial sets model, how the Lack-Paoli nerve can be regarded as an embedding to a "simplicially enriched model", but in the end I can't see why we would want to deal with $(\infty,2)$-categories in the first place.
All I've seen so far is their use in low dimensional TQFT, and also as a way to encode the $(\infty,2)$-category of $(\infty,1)$-categories, both in the context of specific models as well as in $\infty$-cosmological contexts.
So (do we care, and if so) why do we care about $(\infty,2)$-categories?
 A: As you say, a major use of $(\infty,2)$-categories is for organising $(\infty,1)$-categories and similar objects (stable $\infty$-categories, $\infty$-topoi, enriched $\infty$-categories, $\infty$-operads...). The importance of the non-invertible 2-cells is the same as in classical $(2,2)$-category theory: they provide natural internal notions of adjunction, base change mappings, lax functor, lax monoidal functor, Kan extension, and so on. An $(\infty,2)$-category can be used to organise collections of these structures and keep track of coherences between them.
A good illustration of the utility of this is the notion of a "six-functor formalism," which Gaitsgory and Rozenblyum (https://bookstore.ams.org/surv-221/) argued is best captured by a certain symmetric monoidal $(\infty,2)$-functor on an $(\infty,2)$-category of correspondences between derived stacks. We definitely need $\infty$ here because the value of such a functor would be something like the derived $\infty$-category of quasi-coherent or constructible sheaves, and the source may also include some derived/higher objects. We definitely need $2$ because the 2-cells of the category of correspondences encode all kinds of coherences between the six functors (for example, base change 2-cells and the higher associativity of compositions of 2-d grids of base change squares). Even if one ultimately only cares about constructing functors out of correspondences on a 1-category, in practice one still needs its universal property among $(\infty,2)$-categories.
See also my paper https://arxiv.org/abs/2005.10496 for a slightly different take to Gaitsgory-Rozenblyum's.
A: One place where $(\infty,2)$-categories shows up is the geometric Langlands program. (As in David Ben-Zvi's comment, this is again related to the TFT example.) Indeed, local geometric Langlands is often stated as an equivalence of $(\infty,2)$-categories:
$$F:D(G((t)))\operatorname{-mod}\cong\operatorname{QCoh}(\operatorname{LocSys}_{\check{G}}(\overset{\circ}{D}))\operatorname{-mod}.$$
(Actually, this is false as stated - the RHS is modified in the actual conjecture, but it's a little hard to state this modification.) Here $G((t))$ is the loop group, and $\operatorname{LocSys}_{\check{G}}(\overset{\circ}{D})$ is the moduli space of $\check{G}$-local systems on the punctured disk. $D(X)$ and $\operatorname{QCoh}(X)$ denote the derived ($\infty-$) categories of D-modules and quasicoherent sheaves on $X$, respectively. The $(\infty,2)$-categories in question are categories of modules over the respective monoidal $(\infty,1)$-categories.
By the way, an $(\infty,2)$-category is an enormous amount of data, and so an equivalence of $(\infty,2)$-categories is an extremely powerful statement. For any objects $X,Y\in D(G((t)))\operatorname{-mod},$ the above conjecture predicts an equivalence (of ($\infty,1)$-categories) $\operatorname{Hom}(X,Y)\cong\operatorname{Hom}(F(X),F(Y)).$
At this point we know what $F(X)$ should be for quite a few $X,$ which makes it possible to extract a lot of nontrivial equivalences from local geometric Langlands. The simplest example is to take $X=Y=D(\operatorname{Gr}),$ $\operatorname{Gr}$ the affine Grassmannian. In this case $\operatorname{End}_{G((t))}(D(\operatorname{Gr}))$ can be identified with the derived Satake category, i.e., the derived category of $G[[t]]$-equivariant D-modules on $\operatorname{Gr}$, and you recover the derived geometric Satake equivalence.
