My Apple Headphones are not listed as output, how to fix this?

Are these "apple headphones" the AirPods Max , AirPods, AirPods Pro, or EarPods?

With these headphones connected to your Mac, click on the Apple Logo > About This Mac > More Info... > System Report... > Hardware > Sound > Audio > Devices

Do these headphones appear under Devices?

So, the issue is with wireless (Bluetooth) devices connecting to your Mac ... correct? If you use wireless keyboard or mouse, can they connect successfully?

Regardless. Is this a silicon (M-series) or Intel Mac?

Ref: If a Bluetooth headset doesn’t work with Mac - Apple Support