It’s there. you may have to scroll down and find it amongst the list of choices.
But yeah, I’ve encountered oddities with Airplay
Tubi would not work with AP in the control centre until I used the cast function within the Tubi app - and then using AP from the control centre worked. Same behaviour with the Netflix app; however it only works by casting directly from the app and not usimg the control centre.
Using AP from the control centre for Apple Music works.
Screen mirroring did not work until I ‘added’ the control to the control center list.
So I went Homer Simpson and added everything I could until there were no more slots to add extra widgets to. as you can see, it’s crowded - so it would be nice to have colour differentiation for everything you add in.
Mirroring everything seems to work, except for Netflix, which I find I have to cast directly from the app.