One answer to your question "why one would be interested in Lagrangian submanifolds" is to quote Weinstein's Symplectic Creed (from a 1981 article), which says *everything is Lagrangian* --- i.e. (1) most constructions in symplectic geometry have an interpretation involving a Lagrangian, and (2) this is a profitable point of view. For instance, if $M$ is symplectic and $M//G$ is its quotient by the Hamiltonian action of $G$ with moment map $\mu$, there is a Lagrangian $\Lambda$ in $M^- \times M//G$ that remembers $\mu^{-1}\{0\}$ and the quotient map $\mu^{-1}\{0\} \to M//G$. This answer is explained in more detail by Stefan Waldmann in the question you link to.

Another answer, which is maybe more relevant given that you write about moving a construction from one place to another, is that Lagrangian correspondences from $M_0$ to $M_1$, i.e. $L_{01} \subset M_0^- \times M_1 = (M_0\times M_1,(-\omega_M) \oplus \omega_N)$ Lagrangian, is the right notion of "a map from $M_0$ to $M_1$". A more obvious notion of map would be smooth maps $\varphi: M_0 \to M_1$ with $\varphi^*\omega_1 = \omega_0$, but all such $\varphi$ are embeddings of symplectic submanifolds, so there really aren't enough for this to be a good notion of morphism.

Alan Weinstein kicked off this point of view (I think the first reference is the same article from above?). He called it the "symplectic `category'". The reason "category" is in quotes is that there's a notion of composing Lagrangian correspondences, i.e. $$L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{M_1} L_{12}),$$ but the resulting thing may not be a(n immersed) Lagrangian if the intersection defining the fiber product is not transverse.

This point of view fits well with the Fukaya category, as Katrin Wehrheim and Chris Woodward (and Sikimeti Ma'u on one of them) showed in a series of five papers, one of them being this paper.

So, my answer to your question "when do you go hunting for Lagrangian submanifolds?" is: whenever you care about the morphisms between two symplectic manifolds.

** EDIT: ** Since you want a concrete situation, here is an example of a result whose statement doesn't involve Lagrangians but whose proof involves using a Lagrangian correspondence to move from one place to another.

In this amazing paper of Ivan Smith's, Smith proves that the natural representation $$\tilde{\Gamma}_g^{\text{hyp}} \to \pi_0\text{Symp}(Q_0 \cap Q_1)$$ is faithful. Here $Q_0 := Z(\sum \lambda_jz_j^2)$ and $Q_1 := Z(\sum \lambda_jz_j^2)$ are quadric hypersurfaces in $\mathbb{P}^{2g+1}$, and varying the $\lambda_j$'s in $\mathbb{C}$ and parallel-transporting induces an action on $Q_0 \cap Q_1$ of the hyperelliptic mapping class group $\Gamma^{\text{hyp}}_{g,1}$ of once-pointed genus-$g$ curves. $\tilde{\Gamma}_g^{\text{hyp}}$ is a certain non-split extension of $(\mathbb{Z}/2\mathbb{Z})^{2g}$ and $\Gamma_g^{\text{hyp}}$.

This is a corollary of the main theorem, which is an identification of Fukaya categories $$D^\pi\mathcal{F}(\Sigma_g) \simeq D^\pi\mathcal{F}(Q_0^{2g} \cap Q_1^{2g}; 0),$$ where the RHS side is the full subcategory of $L$'s on which quantum multiplication by $c_1(TM)$ acts by zero. The way Smith gets this identification is by embedding both categories into the Fukaya category of the total space $Z := \text{Bl}_{Q_0\cap Q_1}(\mathbb{P}^{2g+1})$ of the pencil generated by $Q_0$ and $Q_1$: $$D^\pi\mathcal{F}(\Sigma_g) \hookrightarrow D^\pi\mathcal{F}(Z) \hookleftarrow D^\pi\mathcal{F}(Q_0^{2g} \cap Q_1^{2g}; 0).$$ Then he identifies the images.

The way he gets the second embedding is by considering a (formal summand of a) Lagrangian $\Lambda$ in $(Q_0\cap Q_1)^- \times Z$, then associating to it an $A_\infty$ functor between the Fukaya categories via Mau--Wehrheim--Woodward's machinery (see YHBKJ's comment on this question). It's easy to describe $\Lambda$: it's the composition of a correspondence $\Lambda_1 \subset (Q_0\cap Q_1)^- \times E$ with $\Lambda_2 \subset E^- \times Z$, where $E$ is the exceptional fiber in $Z$. $E$ is isomorphic to $(Q_0 \cap Q_1) \times \mathbb{P}^1$, so define $\Lambda_1 := \Delta_{Q_0 \cap Q_1} \times S^1_{\text{eq}}$ (so the functor associated to $\Lambda_1$ just multiplies a Lagrangian in $Q_0 \cap Q_1$ by the equator). To get $\Lambda_2$, note that $E$ is the symplectic quotient of $Z$ by $U(1)$ (as long as we're careful about symplectic forms); set $\Lambda_2$ to be the correspondence I mentioned at the very top of my answer.

Sorry, I'm sure there are much less complicated answers to your question. But this is the first example that came to mind of the "symplectic category" approach answering a question that's not phrased in terms of Lagrangians.