airplay

I see. As it happens, the Zoom application does not offer its own AirPlay function. You can still use the general system Screen Mirroring function to display your entire computer screen on your big screen.

You'll find both Screen Mirroring and Audio AirPlay options under Control Centre on Monterey.

What application are you using to play the video?

In Apple's "TV" application, the AirPlay icon appears near the volume slider in the lower left corner, but only after you start playing a video.

In Apple's "Photos" application, the AirPlay icon appears at the right end of the progress bar, again during play.

That's correct. AirPlay is for video and audio. You can generally AirPlay video or audio directly from the player (when supported, i.e. YouTube doesn't support AirPlay because Google), but you can also AirPlay system audio from Control Centre.

What you're wanting is screen mirroring, which is also available under Control Centre.

No idea. I don’t use either of those. You might check their help docs, ask their support staff, or consult other users of those tools.