There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $\mathbb A^{1}$-homotopy theory.

My question is twofold:

Are there useful examples of operads in algebraic geometry?

Is there some intrinsic property of schemes which makes them incompatible with operatic structures? What about (higher) stacks?

For context, my main interest is in the $E_{n}$ operads. For instance, one would naïvely imagine that the moduli space $M_{0,n}$ of genus-zero marked curves could be an algebraic geometric counterpart of the space of little disks, but there seems to be no way to define the operadic composition without passing to the Deligne–Mumford compactification of the moduli space.