I have a couple of BT headphones here. An older Bose QuiteComfort 35 II
and a recent JBL Soundgear Sense.

The JBL works fine with HFP and A2DP profiles and all applications.

The Bose does not. In the Blueman applet, I can see that it has A2DP in
the list of capabilities. I can even play music directly through the
ALSA sink underneath (see below) and it sounds fine. But pavucontrol
does NOT show that in the list of profiles, only CVSD and mSBC are
there.

So in my opinion something is odd in the negotiation between the layers.
Please let me know if you need more information.

aplay -D bluealsa:PROFILE=a2dp /tmp/bla.wav
[276130] D: bluealsa-pcm.c:1473: Getting BlueALSA PCM: PLAYBACK 00:00:00:00:00:00 a2dp
[276130] D: bluealsa-pcm.c:1222: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Setting constraints
Wiedergabe: WAVE '/tmp/bla.wav' : Signed 16 bit Little Endian, Rate: 44100 Hz, stereo
[276130] D: bluealsa-pcm.c:561: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Initializing HW
[276130] D: bluealsa-pcm.c:600: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: FIFO buffer size: 1024 frames
[276130] D: bluealsa-pcm.c:605: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Selected HW buffer: 4 periods x 24000 bytes == 96000 bytes
[276130] D: bluealsa-pcm.c:636: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Initializing SW
[276130] D: bluealsa-pcm.c:636: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Initializing SW
[276130] D: bluealsa-pcm.c:636: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Initializing SW
[276130] D: bluealsa-pcm.c:677: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Prepared
[276130] D: bluealsa-pcm.c:636: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Initializing SW
[276130] D: bluealsa-pcm.c:388: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Starting
[276133] D: bluealsa-pcm.c:231: /org/bluealsa/hci0/dev_28_11_A5_DA_F2_89/a2dpsrc/sink: Starting IO loop: 8

Best regards,
Eduard.