I’ve accomplished this by remote controlling a spare Android phone. It seems to be the only way currently (Since remote controlling Chrome does not allow casting), but it works quite well actually for a work-in-progress. So, we could have this in Home Assistant soon!
I’ve got pull requests pending for pychromecast and HAss which provide basic support for app casting functionality:
https://github.com/home-assistant-libs/pychromecast/pull/359
https://github.com/home-assistant/core/pull/35114
Additionally, Netflix support is provided in my custom component cast_extensions: