A adição de acls para permitir explicitamente o acesso do mpd aos dispositivos de som resolveu o problema:
sudo setfacl -m u:mpd:rw /dev/snd/*
Bevor fazendo isso, os acls eram assim:
# file: dev/snd/controlC0
# owner: root
# group: audio
user::rw-
user:lightdm:rw-
group::rw-
mask::rw-
other::---
# file: dev/snd/pcmC0D0p
# owner: root
# group: audio
user::rw-
user:lightdm:rw-
group::rw-
mask::rw-
other::---
# file: dev/snd/pcmC0D1p
# owner: root
# group: audio
user::rw-
user:lightdm:rw-
group::rw-
mask::rw-
other::---
# file: dev/snd/seq
# owner: root
# group: audio
user::rw-
user:lightdm:rw-
group::rw-
mask::rw-
other::---
# file: dev/snd/timer
# owner: root
# group: audio
user::rw-
user:lightdm:rw-
group::rw-
mask::rw-
other::---
Como o mpd já era membro do grupo de áudio, esses acls deveriam ter sido suficientes. Eu darei a recompensa àquele capaz de explicar o que estava acontecendo aqui.