Do modular forms show up in the cohomology of moduli spaces of unmarked curves? Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + \frac{ \sum_{i=1}^k n_i}{2}$ for positive natural numbers $n_1,\dots, n_k$, then there are several natural maps
$$ \overline{\mathcal M}_{1,n_1} \times \dots \times \overline{\mathcal M}_{1, n_k}  \to \overline{\mathcal M}_{g}$$
one for each perfect matching of the $\sum_{i=1}^k n_i$ marked points that connects the $k$ elliptic curves.

Is the functoriality map $H^i\left(\overline{\mathcal M}_{g}, \mathcal O_{\overline{\mathcal M}_{g}}\right) \to H^i\left( \overline{\mathcal M}_{1,n_1} \times \dots \times \overline{\mathcal M}_{1, n_k}, \mathcal O_{ \overline{\mathcal M}_{1,n_1} \times \dots \times \overline{\mathcal M}_{1, n_k}}\right)$ ever nontrivial for $i>0$?

The motivation is that this would imply that some piece of the motive of $\overline{\mathcal M}_g$ is easy to understand, because it will be a tensor product of the motives of modular forms. You may of course substitute "Hodge structure" or "Galois representation" in for "motive" if you prefer.
I think not because it seems like that would make it "too easy" to understand part of the cohomology, but maybe it's difficult just because the map is sometimes zero and sometimes nonzero and there's no good way of telling when.
 A: Not an answer but a comment. There's a paper by Pikaart, "An orbifold partition of $\overline M_g^n$", in which he proves (among other things) the following result. Consider the boundary stratum in $\overline M_g$ parametrizing a genus one curve and a curve of genus $g-11$ meeting in $11$ nodes. The corresponding pushforward map
$$ H^0(\overline M_{g-11,11}) \times H^{11}(\overline M_{1,11}) \to H^{33}(\overline M_g)$$
is injective for $g$ large enough. This is Corollary 4.7 in his paper. This doesn't answer your question since this gives cohomology of type $(22,11)$ and $(11,22)$, an $11$-fold Tate twist of the motive attached to the cusp form $\Delta$, but morally it seems very similar. 
It might be possible to answer your precise question by reading Pikaart's paper more carefully. Also, I believe (but I never compared the two carefully) that Pikaart's construction was essentially reinvented by Teleman and is described in Section 5 of his paper on the classification of 2d semisimple field theories. In fact, Teleman writes that this section is the key part of the whole argument. You might find Teleman's paper easier reading.
