AirPlay icon does not appear in music toolbar

AirPlay icon (between sound level control and lyrics icons) does not appear in Music.

I have connected Bluetooth speaker and it is showing in sound devices. It can be selected as default sound output device, but this will redirect all system sounds through it as well.

I would like to stream only Music to this bluetooth speaker.

What I have found is that if I open Air M1 in close proximity, then AirPlay Icon appears in Music and then I can of course select my Bluetooth speaker. Once I close the Air M1 and turn off the Bluetooth speaker, there is a message "Selected device is not available" and icon disappears.

Attached is screenshot with properly showing AirPlay icon and bluetooth speaker selected.

This explained behaviour happens in identical way on both my MacBook Pro 2018 and 2019.

Once you start the Bluetooth speaker again and it appears in Bluetooth menu, Music no longer recognizes it as usable device. See screenshots:

And now even that the speaker is connected to computer, it is no longer available for use with Music: