The iPhone, Apple TV box, iPad, iPod touch, and Mac support only Apple’s own AirPlay protocol on a system level. Any other stream sending protocols would have to be included in each third party app, as programmed by their developers. Those other protocols wouldn’t be included in Apple’s own apps, nor system-wide.
Use AirPlay to stream video or mirror the screen of your iPhone or iPad - Apple Support
Prefer in-app AirPlay options over system-wide Screen Mirroring, if available, for best support and quality.
The YouTube app on iPhone, iPad, or iPod touch would have Chromecast included. The YouTube app on non-Apple devices wouldn’t support AirPlay. Streams would have to be AirPlay-compatible; not sure if that excludes some YT videos. Most videos have video and audio in multiple formats, to facilitate compatibility with many devices.
When your Apple TV box is not listed as option, verify that your iPhone is connected to the same home network. Can you discern a pattern for when it doesn’t show?
Roku supports AirPlay receiving for select models. It may not be activated by default.
Select smart TVs support AirPlay receiving. It may not be activated by default.
List of AirPlay-enabled smart TVs and third party streaming devices - Apple
