Airplay/Screen Mirroring in Big Sur

Okay. Here is what I have figured out after spending some time with Big Sur:

1. There are 2 places online to find information about how Airplay works in Big Sur, and both of them have some helpful information:
2. At least with U.S. and Canadian websites, the way to do Airplay now is to look for the Airplay icon in the video itself, instead of working from Screen Mirroring in the menu bar (although it still helps to have it in the menu bar). Click on the Airplay icon in the controls section of the actual video you want to airplay, and it will give you choices of which Apple TV you want to mirror to. Instead of trying to click on the words, you need to click a box to the right of the words, and it should play automatically.
3. I have found that will not work on French websites, and I am assuming thus that it will not work on some other countries' websites. I could not find out why. Although the Airplay icon exists in the European videos, it either does nothing or it actually turns Airplay off instead of turning it on.
4. For European websites, or if the method described above (para 2) does not work elsewhere, it is still possible to get Airplay. Follow these steps: (1) In System Preferences, go to "Displays" and check "Show mirroring options in the menu bar when available." (2) Click on "Airplay Display" and select the Apple TV you want to play to -- Note that this is where the "OFF" option is, as it may become difficult to see it once you optimize the image for your TV; (3) A second box will open up for "Display," and select either "Optimize for Built-inRetina Display - Best for Built-in Retina display (which will give you a full size image on your computer screen to make it easier to exit Airplay) or "Optimize for [the Apple TV you are using, which will give you a better full-screen image on the TV, but might greatly alter the size of everything you see on your computer. (4) Once you optimize for your tv, if you can't see the Airplay Display box or your cursor well enough to turn it off, you can exit Airplay by simply shutting the lid on your laptop. (5) If your computer makes everything super tiny on your computer screen when optimizing for your tv (as it did to me), you can choose to make your cursor larger to help with that by going to System Settings - Accessibility - Cursor, and making your cursor large enough that you can see it to click on "off" when you finish watching.

So Airplay is actually quite easy with Big Sur as long as you are Airplaying from an American or Canadian website to an Apple TV. I have not found a better solution for Airplay from those websites that do not work the way Big Sur intends them to and, for me, that has been every European website I have tried. I suspect it may be that Big Sur is not yet fully functional in all countries, or else portions of it won't work internationally, perhaps as part of Apple's security measures. It is still possible to use Airplay for videos from other countries, but it takes a bit of work as described above.

Yes, I found that too. It is still much less convenient the ca way the screen mirroring button works than having the old Airplay button in the menu bar (no exit option, for one thing). I found a Bug Sur online manual that said the intention is to use the Airplay button in the website’s own video. But I found those have quit working or have been removed over the past few weeks. I’m going to talk to someone at Apple about it to see if there is something I am missing or if it is a problem they need to improve with an update. I’ll report back if I find any info that may help others here.

I forgot to insert the links for the web pages on Airplay above:

https://support.apple.com/guide/macbook-pro/use-airplay-apdf1417128d/mac

https://support.apple.com/en-us/HT204289#mac