Music Player Daemon (MPD) is a server-side application for playing music. Through plugins and libraries it can play a variety of sound files while being controlled by its network protocol. There are many MPD clients and if you want all the features you can imagine in a music player and the lightest solution for this you should give MPD + GMPC a try.
Adding MPD (and thus any MPD client) to the Ubuntu Sound Menu is very easy thanks to Vsevolod Velichko who maintains a PPA for a package he created to add MPD to the Sound Menu.
It will only work if your music is in (or linked to) /var/lib/mpd/music. In other cases it'll work in sound-menu, but it'll break MPRIS.
Vsevolod hopes he'll be able to make it reconnect MPD and be able to restart MPD from the sound menu as well as make the music library configurable but for now these are not available.
Many thanks to WebUpd8 reader Gnaag for the tip and all the info!