AirPlay icon appear/disappears every time I click on "Show mirroring options in the menu bar when available", is that normal? This model Macbook Air Mid 2011 is compatible by hardware with AirPlay. Since I can not see the AirPlay icon at the top bar, I can not connect to SideCar.
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AirPlay mirroring won't work more on Catalina since the SkyLight/CoreDisplay/OpenGL are patched and from Mojave, instead it will work again if you'd not used the "Video Patch", also in a non-video-accelerated Catalina system AirPlay mirroring (to an appleTV for example) should work.
Sidecar not, because apart the AirPlay it requires a Metal GPU.