I lost an hour today troubleshooting why I could not mirror my iMac to AppleTV, as I have done for about 77 years. I can only surmise this new issue popped up as a result of upgrading the OS to BigSur. It would not connect because the audio was connected (not the right word, keep reading) to my headphones.
The last time I used my desktop I was listening to music through a pair of bluetooth headphones. After I was finished, I powered down the headphones and put them away. Fast forward to today, I jumped through hoops that included, restarting both devices, double-checking all restrictions and permissions, reading no fewer than 9 articles (none helped), and some other things. Finally, I happened to notice in the new control center that the audio section had a headphones icon. I clicked on it and saw AppleTV as a choice. I chose AppleTV and then tried to mirror it again. That did it.
For those of you who mirror to an AppleTV, if it does not connect, check to see if your audio is connected to the AppleTV. Seems a silly reason to fail an operation rather than flash a warning/reminder, but that is the case.