You don't get anything just from knowing that a non-elementary hyperbolic quotient exists. However, the problem of constructing such quotients of mapping class groups appears very difficult, and can be viewed as a step on the way to constructing other interesting classes of quotients that you might hope to study, like finite quotients or virtually abelian quotients.
For instance, here are two important open problems about the finite-index subgroups of mapping class groups (let's say for closed surfaces $S$ of genus $\geq 3$, to be safe). The first is easy to understand.
Virtual first betti numbers: Does $\mathrm{Mod}(S)$ have a finite index subgroup that surjects $\mathbb{Z}$?
The second is morally asking: "Does every finite-index subgroup of $\mathrm{Mod}(S)$ come from a finite-sheeted cover of $S$?"
Congruence subgroup property: Does every finite-index subgroup of $\mathrm{Mod}(S)$ contain a principal congruence subgroup, i.e. the kernel of a natural map $\mathrm{Mod}(S)\to\mathrm{Out}(Q)$, where $Q$ is a finite characteristic quotient of $\pi_1(S)$?
I'll take for granted that these are interesting questions, but could explain more if necessary. The second one is certainly related to the profinite rigidity questions mentioned in the question. (Added:) In any case, even if you don't like either of these questions specifically, the point is that we are interested in, and don't know much about, the finite and virtually abelian quotients of mapping class groups.
Note that either of these questions can be answered by finding suitable quotients of $\mathrm{Mod}(S)$: the first (positively) by finding a virtually abelian quotient, the second (negatively) by finding a finite quotient that doesn't factor through some $\mathrm{Out}(Q)$.
Neither can be answered directly by a non-elementary hyperbolic quotient. However, we know a lot about hyperbolic groups, and so if we can construct hyperbolic quotients, we might hope to go further and construct hyperbolic quotients with a rich supply of finite quotients (for instance if they're residually finite), and thereby answer these questions.
There's some precedent for this hope. The Virtual Haken conjecture and its cousins for hyperbolic 3-manifolds asked for rich supplies of finite quotients of hyperbolic 3-manifold groups. One of the crucial tools in Agol's proof turned out to be Wise's Malnormal special quotient theorem, which guarantees that hyperbolic 3-manifold groups have non-elementary hyperbolic quotients which are themselves residually finite. (This is a slightly backwards telling of the Virtual Haken story, but the point is that these are the tools you need.)
Indeed, in some ways the starting point for the proof of the Virtual Haken conjecture was a paper of Agol, Groves and Manning, which proved it (modulo the work of Kahn--Markovic) under the hypothesis that all hyperbolic groups are residually finite. I view the work of Behrstock, Hagen, Martin and Sisto as the analogue of the Agol--Groves--Manning paper in this story.
