I also have the same problem since upgrading to MacOS Big Sur 11.6.4
I have done all the firewall suggestions and have the screen mirroring accessed. There's simply no option for me to turn on AirPlay. The only option is Off. The problem is with MacOS Big Sur, not with any of the technologies to which I am trying to connect.
Apple, you need to fix this. I spent hours on the phone and I am not going to erase my hard drive and start all over at the AppleStore. Someone has to be something less of a genius who can figure this out.
ruthegoldstein
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