Danke Gerhard! Das hat zwar leider nicht funktioniert, aber dein Hinweis hat mich auf die Idee gebracht, auf der von dir auch gelinkten Seite der Apple Support Community ein bisschen weiter herumzulesen. Und da findet sich die Lösung auch für mein Problem (und für das von diesem und jenem):
»Anyone suffering from this issue, please do the following:
 
1. Open up System Information (located in /Applications/Utilities, or option+click the apple logo in the menu bar and select it).
2. On the left side bar locate "Extensions" (you'll have to scroll down a bit)
3. Look in the list of extensions for "APExtFrameBuffer" as shown in the screenshot.
 
Screen Shot 2014-03-01 at 11.58.56 AM.png
 
Also look for "AirParrotDriver". I had two of them listed, one in /System/Library/Extensions and one in /Library/Extensions.
 
Screen Shot 2014-03-01 at 12.13.56 PM.png
 
4. If any of these items are found, open Terminal (in Applications>Utilities) and copy/paste these (respectively):
 
sudo rm -rf /System/Library/Extensions/APExtFramebuffer.kext
sudo rm -rf /System/Library/Extensions/AirParrotDriver.kext
sudo rm -rf /Library/Extensions/APExtFramebuffer.kext
sudo rm -rf /Library/Extensions/AirParrotDriver.kext
 
If they yield an error, then the file probably doesn't exist. Any one of these could cause the issue, so make sure they are all deleted.
 
5. Enter your administrative password when prompted.
6. Restart your computer.«
Jetzt lässt sich im Pulldown Menü AirPlay Monitor das Untermenü "Spiegeln" wieder anwählen. 
Udemo