I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single element of this group other than complex conjugation; the only thing you can do is to study it indirectly via its representations, like the cyclotomic characters and the $\ell$-adic representations attached to modular forms.

Similarly one of the motivations for studying dessin d'enfants that I've heard stated in seminar talks etc. is that since the absolute Galois group acts faithfully on the set of all dessins, we can learn things about the absolute Galois group by studying its action on this set. Just as an example I quote:

Fortunately, Belyi's Theorem provides us with an explicit realisation of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ in terms of bipartite maps, which is beginning to add to our rather meagre knowledge of this complicated group.

My question is: just what kind of Galois-theoretical or number-theoretical statements do people hope to prove by studying dessins? Are there at this point any theorems proven via dessins which can be stated without using the language of dessins?