I cannot find airplay on my 14" pro laptop running monterey

Sorry, my fault.

I airplay to my TV, but currently I'm on Ventura. Here are the docs for Monterey:

Use AirPlay to stream what’s on your Mac to an HDTV - Apple Support

Here is how to airplay to Mac from iOS 16 device:

Wirelessly stream video, photos, and audio from your iPhone to a Mac - Apple Support

You can also stream from the Music app to TV, HomePod, or other Airplay speaker. Ironically the icon for this is the same as AirDrop on the Music window.

The key to using AirPlay is to make sure both devices are using the same wifi network. There is no place in settings to manage the AirPlay devices or even turn it on, they should just show up when you need them. The only settings you may have are for receiving AirPlay on your Mac.

Let me know if you are doing a specific task and it is not working. Some TV's are not compatible or require changing settings on the TV to make it possible.

Open a Finder window. If Airdrop is not in the sidebar, then go to Finder > Settings > Sidebar. Make sure Airdrop is selected. If it was not selected, you should now have it in the sidebar of a Finder window. Select Airdrop and drag a file to a person in that window.

If no one shows up in that window, there are other things to try such as, make sure destination device is unlocked, connected to Wifi, connected to Bluetooth, or allowed to receive from everyone or only those in their Contacts.