AirPlay code

What is your model Apple TV? [1] [2] What is its version system software (tvOS)?

What is the version macOS on your iMac? [3]

• AirPlay would ask for a 4-digit pairing code when no AirPlay password is set. The code would be shown on the TV screen (interrupting anything else that was playing at that time).
• AirPlay would ask for a 6-character (or more*) password when a AirPlay password is set. This code can be set in Settings on Apple TV (as the receiving device). This wouldn’t show a code on screen, and viewers wouldn’t be interrupted by possible failed attempts from unauthorized people. *Toggling off-on would auto generate a new password, but a custom password can be set, if you wish.
• AirPlay has an off-on switch on the receiving device. Verify that it is set to ‘On’.
• AirPlay has access rules, of who is allowed to stream. I would recommend Anyone on the Same Network in most situations.

Use AirPlay to stream audio and video on Apple TV - Apple Support

Some content may not work for AirPlay, as set by the content provider or app.

- DVD-Video as played in the Apple DVD Player app prohibits AirPlay screen mirroring and capturing content.

- Web video may have DRM properties that prohibits AirPlay (e.g. Netflix).

- Web video may have video properties that are not compatible with basic AirPlay. Your Mac has (slightly) different playback abilities than the Apple TV box. It could work differently for screen mirroring, though, which can be seen as capturing and re-encoding.

- Content may have minimum system requirements for the receiving device (e.g. AirPlay 2 compatibility). That could exclude legacy models Apple TV for some content.

So it depends somewhat on what you are trying to stream. Can you tell a bit about that?

The AirPlay button in the controller bar of Safari web video is functional for AirPlay. If you downloaded and played back through QuickTime Player, then that would also have an AirPlay button in the controller bar.

Vimeo does have an app for the Apple TV box, so you could play it without going through the Mac.

If the sending device and the receiving device are in separate rooms, then that could make it very inconvenient for controlling the AirPlay stream. Using the Apple TV remote in any way (except audio controls) would abort the AirPlay stream.

Instead of AirPlay, consider Home Sharing for downloaded files. Instead of pushing content to other devices, pull content from a networked computer while using Apple TV. Use Home Sharing to share media from a computer to other devices - Apple Support.

Apple Home Sharing does depend on comp,atible media files (.mp4, .m4v, .mov, m4a, mp3, .jpg, tif, …) in the libraries of the TV app, Music app. For other file formats, you may need to convert, or use a different local sharing framework (VLC, Plex, Infuse, …).