I figured the argument would be something like that. Does this also hold for surface groups though? If I understand correctly, you use that to get goodness of the mapping class group of $S\setminus\lbrace x\rbrace$ from Lemma 3.3 when S is closed and has genus greater than 0.

Thank you for this very detailed comment. I've been trying to flesh out your argument for goodness of the closed mapping class group implying goodness for the punctured mapping class group. The one thing I can't figure out is why $\pi_1(S_{g,n})$ satisfies the extra conditions on $N$ in Lemma 3.3 of the paper you linked. Is there just some general theorem I'm missing for when the cohomology groups $H^q(N,M)$ are finite for finite $M$?

@DanRamras Thanks for the link! I had a read through the appendix and I'm not entirely sure it's helpful for my purposes. It does construct a similar operad in which gluing instructions do not seem to be part of the objects, but they also don't go through a lot of the quotient constructions in Tillmann's original paper.