Common causes for the menu icon not showing:
- Network configuration incorrect -- target device [e.g. Apple TV] for AIrPlay and Mac must be on same network
- Need WiFi turned on for both the Mac and the Apple TV, or they must be connected to the same router
- WiFi interference
- No target device [e.g. Apple TV] to AirPlay to is turned on nor visible/detectable
- Apple TV is on network but AirPlay is not enabled on the Apple TV
- Need to turn on "Automatically allow signed software to receive incoming connections" in Firewall
- Mangled or corrupt MacOS install or AirPlay TVOS install
If there is no AppleTV or AirPlay capable device on your network or properly configured on your network, or your OS is not installed properly (or maybe if you have made custom changes to the OS like the time you boasted about removing Safari as you once indicated you had done -- was that you?), it might not show up.