airplay
Apple hardware requirements state that AirPlay is not supported on my 2010 Macbook pro.
However, I can use AirPlay to stream music/movies from iTunes to my Apple TV (4th gen), as well as YouTube videos on Safari.
So why can't I get a menu bar app? AirPlay clearly is supported.
It's not just for AirPlay. It's general-purpose media redirection. You can use it to switch between the built-in earpiece, the speaker on the bottom, Bluetooth earpiece, etc when on a phone call. For example, I placed a call then went over to the audio controls in the multitasking bar.
It just so happens that the primary use is for AirPlay and media playback.
You can mirror your Mac's screen, Apple calls it AirPlay Mirroring. The section on the Apple TV page you mention actually refers to streaming your "photos, home videos, and music collection", which you can do via iTunes on a Mac or PC.
Per Apple's support article on AirPlay Mirroring, to use it you need:
Apple doesn't provide a means for Windows PCs to do AirPlay mirroring, but you might be able to find a third-party solution.
The Apple TV and the Mac need to be on the same network, one will not be automatically created, but you should be able to create a Wi-Fi network from your Mac and have the Apple TV join it if you don't have a network already set up.
The quality is not the same as if you just plugged your Mac in via HDMI, it transcodes your desktop to a video format the Apple TV can read (H.264), since sending uncompressed video would require more bandwidth than any home network could provide. Quality and performance can depend on your Mac and network speed. I haven't tested it thoroughly, but I suspect it wouldn't be appropriate for gaming, as there would be latency issues.
Best Answer
AirPlay is supported, extending or mirroring your Mac's desktop over AirPlay is not supported. As far as I know, the only way to stream from the menu bar is to stream system audio by Option-clicking the sound/volume icon. i think video streaming is still per-app if you can't do AirPlay desktop mirroring/extending.