Google Chromecast Audio.

[ChrisPa]

That's exactly how it works. The only time the Chromecast will stream from the phone is when the content is stored locally on the phone.

Or if (on android) you use the cast option in the settings at the top of the screen. In this mode the phone is using the Chromecast as a WiFi speaker.

The apps need to be Chromecast aware - eg Spotify, BBC iPlayer radio - for the Chromecast to natively play the stream.