Bluetooth headsets have 2 modes of operation.
Mode 1: high quality for listening to music
Mode 2: low quality for voice communications
Mode 1 is the default. As soon as you switch your headset on, it is in this mode and you're able to listen to your media player/streaming website in high quality.
Mode 2 is activated as soon as some app activates the headset microphone. The headset switches to voip operation, and this is much lower listening quality than mode 1. It switches back to mode 1 as soon as you close that app or as soon as that app deactivates the headset microphone.
OBS will trigger this switch to mode 2 as soon as it is started, if you configured the headset microphone somewhere in OBS. To prevent this switch, you have to remove the microphone from OBS Settings->Audio (including "default" device) as well as from every audio input source you might have created as source in some scene. If you did this, OBS will not initialize the microphone any more, and the bluetooth headset will remain in high quality music mode.
On the other hand, if you need the microphone, because you participate in some kind of voice chat, and you want to transmit this chat with OBS, you have no choice but live with the reduced music quality of the headset.
This is bluetooth-specific behavior and not the fault of OBS. The low quality mode has something to do with power saving, bluetooth bandwidth and latency. Cabled headsets don't behave like this, and there are also wireless headsets who are able to circumvent this bluetooth behavior by optionally using some kind of proprietary receiver dongle instead of native bluetooth connectivity. There are also wireless headsets who can be optionally connected with a cable, and while connected with a cable they behave like a cabled headset without this mode switching. This all depends on the model and features of your headset.
Because of this weird behavior of wireless headsets, every streamer has probably a drawer full of unused bad-bought headsets.