Did a new fresh install and installed only the Bluetooth Audio Changer device. When Kodi is running I don't get any audio output at all, but if I ssh into the device and run systemctl stop kodi and pactl load-module module-udev-detect I suddenly can start streaming music again from my Android phone.
So I can at least confirm that streaming works, it's not a HW related problem, and thus I just have to assume it's something that's become broken in either LibreELEC or the add-on itself with an update at some point.
Code
LibreELEC:~ # pactl list short sinks
1 alsa_output.0.iec958-stereo module-alsa-card.c s16le 2ch 44100Hz RUNNING
LibreELEC:~ # pactl list short sources
4 alsa_output.0.iec958-stereo.monitor module-alsa-card.c s16le 2ch 44100Hz IDLE
5 alsa_input.0.iec958-stereo module-alsa-card.c s16le 2ch 44100Hz SUSPENDED
7 librespot_sink.monitor module-null-sink.c s16le 2ch 44100Hz SUSPENDED
12 bluez_source.14_9F_3C_81_30_CC.a2dp_source module-bluez5-device.c s16le 2ch 44100Hz RUNNING
It'd be great if someone from the developer team could look into this.