I have issues too when attempting screen mirroring on my MacBook Air 2020 M1 (16GB 8 Core) via AirPlay to an Apple TV (Apple TV is called "Living Room" for AirPlay).
Firstly, I was not seeing any Apple TVs as options under Display, even when forcing Display options to always appear in Menu Bars... interestingly enough, I could see my iPad Pro 13" as an option.
I had to revert to initiating a buggy AirPlay mirroring session by going to System Preferences > Display > AirPlay Display and then I could see my Apple TVs. I initiated it but then found audio wasn't going over, even when I forced it via Audio options once the screen share had initiated.
Also, although the screen sharing worked, it was buggy when doing a full screen Zoom, where the display repeated a distorted strip of the Zoom video-share at bottom 10% of image).
Audio normally goes over when I use AirPlay mirroring on my MacBook 16" 2019.