Bluetooth headphones don't work at all :(

On Elementary OS Hera 5.1.7 (based on 18.04.4) using kernel 5.8.0-39 (necessary for hardware compatibility reasons), I was able to resolve this issue by

1. installing pavucontrol and blueman and
2. editing /etc/bluetooth/main.conf "MultiProfile=multiple" under "general" (see https://wiki.archlinux.org/index.php/Bluetooth_headset#A2DP_not_working_with_PulseAudio) change "AutoEnable=false" (not sure if that did anything)

Before this, I couldn't connect to bluetooth audio sinks, period, unless I used bluetoothctl. At that point, it would only connect to the duplex low quality sink (HSP/HFP) and not A2DP (high audio quality).

Now it appears to be able to work well enough in either mode, with mode-switching on the fly, however, with consistent errors:

[  994.225669] Bluetooth: hci0: urb 0000000085e83e50 submission failed (90)
[  994.225675] Bluetooth: hci0: sending frame failed (-90)
[  994.238145] Bluetooth: hci0: SCO packet for unknown connection handle 0
[  994.238153] Bluetooth: hci0: SCO packet for unknown connection handle 0
[  994.238155] Bluetooth: hci0: SCO packet for unknown connection handle 0
[  994.238157] Bluetooth: hci0: SCO packet for unknown connection handle 0

Hi I had a similar problem. (The Device showing connected but not appearing in the sound settings and then disconnecting(additional problem))

I referred to this answer and the solution sudo apt-get install blueman worked for me. Figured out that it might help you out.

Note to Mods: This is not an answer but I don't have enough reps to comment, hence you may add it as comment and remove it.

It is a possibility the bluetooth adapter just isn't supported in your current kernel. I'd recommend upgrading with the ubuntu HWE, and then making sure the "linux-firmware" package is installed.

Run the following in a terminal:

 sudo apt-get install --install-recommends linux-generic-hwe-18.04 xserver-xorg-hwe-18.04 linux-firmware

Then reboot and test if it works.