I suspect you need to change the audio output setting.
Unfortunately I don’t know about 12.4 yet - I’m still on 10.13. There, I’m talking about System Preferences > Sound > Output > “Select a device for sound output”. That should allow the Bluetooth headphones to be selected from a list of possible outputs.
Normally, that setting controls all sound from the system. If you would like to, for example, direct Safari’s output to the Bluetooth headphones, but have other sounds still directed to the system speakers, have a look at Rogue Amoeba’s SoundSource app.
I’m not sure why the headphones didn’t show there before - I don’t use Bluetooth cans very often. Perhaps they need to be in some particular powered on / paired / active state to show up? Anyway, glad you have it working.