The braid groups are good (which are mapping class groups of punctured disks) by Proposition 3.5 of
Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A., Cohomological goodness and the profinite completion of Bianchi groups., Duke Math. J. 144, No. 1, 53-72 (2008). ZBL1194.20029. (You can find a version here but missing the statement of this Proposition.)
The summary of the proof is that if $G$ is good, and $H$ is commensurable with $G$, then $H$ is also good (Lemma 3.2). Hence we may pass to the pure braid group $P_n$. This is an extension of $P_{n-1}$ by $F_{n-1}$ (essentially the Birman exact sequence). Hence by induction, $P_n$ is good using Lemma 3.3 (residually finite extensions of good groups by certain good groups such as finitely generated free groups are also good).
By a similar inductive argument, the mapping class groups of the $n$-punctured sphere and torus are good. Moreover, the mapping class group of the genus $2$ surface is a central extension of the mapping class group of the $6$-punctured sphere by $\mathbb{Z}/2$ (generated by the hyperelliptic involution). Hence by Lemma 3.3 the mapping class group of the genus $2$ surface (and also with $n$ punctures by induction) is good.