The latest package for MPD v0.22.3, buster backport 0.22.3-1~bpo10+1 is bugged. The process cannot write the PID file to the location defined in the config.
The bug is due to a missing systemd tempfile config I failed to ship in the package when I backported 0.22.3 to Debian/Buster.
A fix is available in
apt update # check the package is available, should be 0.22.3-1~bpo10+2 apt policy mpd # install apt install mpd=0.22.3-1~bpo10+2
Alternatively you can download the package and run
apt install ./<package>.deb :
Special thanks to the user who reported the issue :) and sorry for the inconvenience.