apple tv app shows black screen when sharing starts

I have a 2019 Mac Mini. I was hoping to rent a movie on it via the Apple TV app, and then watch that via screensharing with my remote family. However, as soon as I turn on any screensharing service (e.g., Kast, Facebook Messenger, Zoom), the Apple TV window showing the movie goes black.

This happens only with Apple TV. I can share other streaming video fine.

Is Apple TV purposefully set up to prevent such sharing, or is this some (hopefully fixable) bug?